This section of Chapter 1 has seven verses:

2 Timothy 1:1-7
1 Paul, an apostle of Jesus Christ by the will of God, according to the promise of life which is in Christ Jesus,
 2 To Timothy, my dearly beloved son: Grace, mercy, and peace, from God the Father and Christ Jesus our Lord.
 3 I thank God, whom I serve from my forefathers with pure conscience, that without ceasing I have remembrance of thee in my prayers night and day;
 4 Greatly desiring to see thee, being mindful of thy tears, that I may be filled with joy;
 5 When I call to remembrance the unfeigned faith that is in thee, which dwelt first in thy grandmother Lois, and thy mother Eunice; and I am persuaded that in thee also.
 6 Wherefore I put thee in remembrance that thou stir up the gift of God, which is in thee by the putting on of my hands.
 7 For God hath not given us the spirit of fear; but of power, and of love, and of a sound mind.

Let us begin with the Barclay commentary

AN APOSTLE’S GLORY AND AN APOSTLE’S PRIVILEGE

First, the paraphrase of the verses:

2 Timothy 1:1–7
This is a letter from Paul, who was made an apostle of Christ Jesus by the will of God, and whose apostleship was designed to make known to all men God’s promise of real life in Christ Jesus, to Timothy his own beloved child. Grace, mercy and peace be to you from God, the Father, and from Christ Jesus, our Lord.

I thank God, whom I serve with a clear conscience, as my forefathers did before me, for all that you are to me, just as in my prayers I never cease to remember you, for, remembering your tears when we parted, I never cease to yearn to see you, that I may be filled with joy. And I thank God that I have received a fresh reminder of that sincere faith which is in you, a faith of the same kind as first dwelt in your grandmother Lois and in your mother Eunice, and which, I am convinced, dwells in you too. That is why I send you this reminder to keep at white heat the gift that is in you and which came to you through the laying of my hands upon you; for God did not give us the spirit of craven fear, but of power and love and self-discipline.

WHEN Paul speaks of his own apostleship, there are always certain unmistakable notes in his voice. To Paul, his apostleship always meant certain things.

(1) His apostleship was an honour. He was chosen for it by the will of God. All Christians must regard themselves as God-chosen men and women.

(2) His apostleship was a responsibility. God chose him because he wanted to do something with him. He wanted to make him the instrument by which the tidings of new life went out to all people. Christians are never chosen entirely for their own sake, but for what they can do for others.  Christians are people who are lost in wonder, love and praise at what God has done for them and aflame with eagerness to tell others what God can do for them too.

(3) His apostleship was a privilege. It is most significant to see what Paul believed it his duty to bring to others – God’s promise, not his threat. To Paul, Christianity was not the threat of damnation; it was the good news of salvation.  It is worth remembering that the greatest evangelist and missionary the world has ever seen was out not to terrify people by shaking them over the flames of hellfire but to move them to astonished submission at the sight of the love of God. The driving force of his gospel was love, not fear.

As always when he speaks to Timothy, there is a warmth of loving affection in Paul’s voice. ‘My beloved child’, he calls him. Timothy was his child in the faith. Timothy’s parents had given him physical life, but it was Paul who gave him [instruction to] eternal life. Many people who never knew physical parenthood have had the joy and privilege of being a father or a mother in the faith.

PAUL’S object in writing is to inspire and strengthen Timothy for his task in Ephesus. Timothy was young, and he had a hard task in battling against the heresies and the infections that were bound to threaten the Church. So, in order to keep his courage high and his effort strenuous, Paul reminds Timothy of certain things.

(1) He reminds him of his own confidence in him. There is no greater inspiration than to feel that someone believes in us. An appeal to the best in someone is always more effective than a threat of punishment. The fear of letting down those who love us is a sobering thing.

(2) He reminds him of his family tradition. Timothy was walking in a fine heritage; and, if he failed, not only would he damage his own reputation but he would lessen the honour of his family name as well. A fine parentage is one of the greatest gifts anyone can have. It is something to thank God for and should never be dishonored. [See Deuteronomy 27:16]

Quoted verse:
Deuteronomy 27:16
Cursed be he that setteth light by his father or his mother. And all the people shall say, Amen.

(3) He reminds him of his setting apart for office and of the gift which was conferred upon him. Once we enter upon the service of any association with a tradition, anything that we do affects not only us. We do not do it in our own strength.  There is the strength of a tradition to draw upon and the honour of a tradition to preserve. That is especially true of the Church.  Those who serve it have its honour in their hands; those who serve it are strengthened by the consciousness of the [spiritual fellowship] of all the saints.

(4) He reminds him of the qualities which should characterize the Christian teacher. These, as Paul at that moment saw them, were four.

(a) There was courage. It was not cowardly fear but courage that Christian service should bring. It always takes courage to be a Christian, and that courage comes from the continual consciousness of the presence of Christ.

(b) There was power. In true Christians, there is the power to cope, the power to shoulder the backbreaking task, the power to stand firm when faced with the shattering situation, the power to retain faith when confronted by the soul-destroying sorrow and the wounding disappointment.  Christians are characteristically people who could pass the breaking point and not break.

(c) There was love. In Timothy’s case, this was love for the brothers and sisters, for the congregation of the people of Christ over whom he was set. It is precisely that love which gives Christian pastors other qualities. They must love their people so much that they will never find any toil too great to undertake for them or any situation threatening enough to daunt them. No one should ever enter the ministry of the Church without a deep love for Christ’s people.

(d) There was self-discipline. The word is so¯phronismos, one of these great untranslatable Greek words. It has been defined as ‘the sanity of saintliness’. In his book on The Pastorals, Sir Robert Falconer defines it as ‘control of oneself in face of panic or of passion’. It is Christ alone who can give us that command of self which will keep us both from being swept away and from running away.  No one can ever rule others without having complete self-control.  So¯phronismos is that divinely given control of self which makes people great rulers of others because they are first of all the servants of Christ and in complete control of themselves [by the power of the Holy Spirit]. ~Barclay Commentary

Now to the rest of the commentaries. We will begin with one of the general commentaries. This one from the F. B. Meyer commentary which covers the first 11 verses. Since the commentary is going to reference each of these verses, I will give them all upfront.

“Stir Up the Gift Which Is in Thee” - 2 Timothy 1:1-11

2 Timothy 1:1-11
1 Paul, an apostle of Jesus Christ by the will of God, according to the promise of life which is in Christ Jesus,
2 To Timothy, my dearly beloved son: Grace, mercy, and peace, from God the Father and Christ Jesus our Lord.
3 I thank God, whom I serve from my forefathers with pure conscience, that without ceasing I have remembrance of thee in my prayers night and day;
4 Greatly desiring to see thee, being mindful of thy tears, that I may be filled with joy;
5 When I call to remembrance the unfeigned faith that is in thee, which dwelt first in thy grandmother Lois, and thy mother Eunice; and I am persuaded that in thee also.
6 Wherefore I put thee in remembrance that thou stir up the gift of God, which is in thee by the putting on of my hands.
7 For God hath not given us the spirit of fear; but of power, and of love, and of a sound mind.
8 Be not thou therefore ashamed of the testimony of our Lord, nor of me his prisoner: but be thou partaker of the afflictions of the gospel according to the power of God;
9 Who hath saved us, and called us with an holy calling, not according to our works, but according to his own purpose and grace, which was given us in Christ Jesus before the world began,
10 But is now made manifest by the appearing of our Saviour Jesus Christ, who hath abolished death, and hath brought life and immortality to light through the gospel:
11 Whereunto I am appointed a preacher, and an apostle, and a teacher of the Gentiles.

Lonely and facing death the Apostle fell back on the bedrock of the will of God. If it were the divine plan that he should finish his life-work in that miserable plight, he was content that it should be so. But he longs to see his beloved son in the faith once more. He desires to stir up the dead coal of his ardor, in which there was fire and heat, but not enough flame.

Apparently the young evangelist was becoming daunted by the gathering difficulties of the time and so Paul sets himself to encourage him. With this purpose in view he adduces his own example, 2 Timothy 1:3, his fervent affection, 2 Timothy 1:4, the memory of the sainted dead, 2 Timothy 1:5, the solemn vows by which Timothy had bound himself at his ordination, 2 Timothy 1:6, the divine donation of grace and power and love, 2 Timothy 1:8, the eternal purpose which had received its fruition in the advent of Jesus, 2 Timothy 1:9, the clear light which His resurrection had thrown on death and the hereafter, 2 Timothy 1:10. Surely such a chain of arguments must have proved irresistible! God’s soldiers must be brave and unflinching in meeting the opposition of the world. When once we realize that the stores which reside in God are at the disposal of our faith, we, too, shall be invulnerable and irresistible. ~F. B. Meyer

Now to the Matthew Henry Concise which covers the first five verses.

2 Timothy 1:1-5
The promise of eternal life to believers in Christ Jesus, is the leading subject of ministers who are employed according to the will of God. The blessings here named, are the best we can ask for our beloved friends, that they may have peace with God the Father and Christ Jesus our Lord. Whatever good we do, God must have the glory. True believers have in every age the same religion as to substance. Their faith is unfeigned; it will stand the trial, and it dwells in them as a living principle. Thus pious women may take encouragement from the success of Lois and Eunice with Timothy, who proved so excellent and useful a minister. Some of the most worthy and valuable ministers the church of Christ has been favoured with, have had to bless God for early religious impressions made upon their minds by the teaching of their mothers or other female relatives. ~Matthew Henry Concise

Now to the specific commentaries.

The verse is generally discussed in three parts:

1] Greatly desiring to see thee.
2] Being mindful of thy tears.
3] That I may be filled with joy.

1] Greatly desiring to see thee.

Greatly desiring to see thee; - Greek, “with yearning as for one much missed.” ~Jamieson, Fausset, Brown

Greatly desiring to see thee; - see 2 Timothy 4:9, 2 Timothy 4:21. It was probably on, account of this earnest desire that this Epistle was written. He wished to see him, not only on account of the warm friendship which he had for him, but because he would be useful to him in his present circumstances. ~Barnes Notes

Quoted verses:
2 Timothy 4:9
Do thy diligence to come shortly unto me:

2 Timothy 4:21
Do thy diligence to come before winter. Eubulus greeteth thee, and Pudens, and Linus, and Claudia, and all the brethren.

Greatly desiring to see thee - In his former epistle he had desired him to stay at Ephesus, there being some work for him to do, which made it necessary he should continue; but now having answered the apostle's purpose, and he standing in need of him at Rome, being without any assistant there, some having left him, and others were left by him in other places, and others were sent by him elsewhere; and it having been some time since he saw Timothy, he longed for a sight of him: ~John Gill

Greatly desiring to see thee.- There is something pathetic in this language. The lonely prisoner calls to him the tears of Timothy at their last parting, and feels a yearning desire to see and counsel him face to face once more. ~People's New Testament

2] Being mindful of thy tears.

Being mindful of thy tears - Perhaps frequently shed, as well as at the apostle's last parting with him. ~John Wesley Explanatory Notes

Being mindful of thy tears - not only at our parting, but also often when under pious feelings. ~Jamieson, Fausset, Brown

Being mindful of thy tears - Alluding probably to the tears which he shed at parting from him. The occasion to which he refers is not mentioned; but nothing is more probable than that Timothy would weep when separated from such a father and friend. It is not wrong thus to weep, for religion is not intended to make us stoics or savages. ~Barnes Notes

Now this from the Biblical Illustrator:

Being mindful of thy tears.
Tears
He seems not merely to speak of the former tears of Timothy shed at bidding Paul farewell (for tears are usually elicited at parting, comp. Acts 20:37), but of his habitual tears under the influence of pious feeling. In this respect also he had him like-minded (Philippians 2:20) with himself. Tears, the flower of the heart, indicate either the greatest hypocrisy or the utmost sincerity. (J. A. Bengel.)

Quoted verses:
Acts 20:37-38
37 And they all wept sore, and fell on Paul's neck, and kissed him.
38 Sorrowing most of all for the words which he spake, that they should see his face no more. And they accompanied him unto the ship.

Philippians 2:20
For I have no man likeminded, who will naturally care for your state.

The power of tears
There is no power that man can wield so mighty as that of genuine tears. The eloquence of words is powerful, but the eloquence of tears is far more so. What manly heart has not been often arrested by the genuine sobs of even some poor child in the streets. A child’s tear in the crowded thoroughfare has often arrested the busy merchant in his hurried career Coriolanus, who defied “all the swords in Italy and her confederate states,” fell prostrate before the tears of his mother: “Oh, my mother, thou hast saved Rome, but lost thy son.” (D. Thomas, D. D.)

Tears described
Tears have been described as the blood of the wounds of the soul, the leaves of the plant of sorrow, the hail and rain of life’s winter, the safety-valves of the heart when too much pressure is laid on, the vent of anguish-showers blown up by the tempests of the soul. ~Biblical Illustrator

Being mindful of thy tears - Whether the apostle refers to the affecting parting with the Ephesian Church, mentioned Acts 20:37 [discussed above], or to the deep impressions made on Timothy’s heart when he instructed him in the doctrine of Christ crucified, or to some interview between themselves, it is not certainly known. The mention of this by the apostle is no small proof of his most affectionate regards for Timothy, whom he appears to have loved as a father loves his only son. ~Adam Clarke

Being mindful of thy tears - shed either at the afflictions and sufferings of the apostle, of which Timothy, being his companion, was an eyewitness, and he being of a truly Christian sympathizing spirit, wept with those that wept; or at their parting from each other, as in Acts 20:37 [discussed above] ~John Gill

Being mindful of thy tears - The words clearly refer to their last parting, probably that referred to in 1 Timothy 1:3. There, with his mind dwelling on the duties to which Timothy had been called, it was natural not to refer to the personal emotions of that parting. Now that absence had increased the yearning desire to see him once again, and so ‘be filled with joy,’ it was as natural to dwell on it. It is characteristic of Timothy’s sensitive, emotional nature that the tears were shed by him. ~Popular commentary

Quoted verse
1 Timothy 1:3
As I besought thee to abide still at Ephesus, when I went into Macedonia, that thou mightest charge some that they teach no other doctrine,

3] That I may be filled with joy.

That I may be filled with joy - By seeing you again. It is easy to imagine what joy it would give Paul, then a prisoner, and forsaken by nearly all his friends, and about to die, to see a friend whom he loved as he did this young man. Learn hence, that there may be very pure and warm friendship between an old and young man, and that the warmth of true friendship is not diminished by the near prospect of death. ~Barnes Notes

That I may be filled with joy - at the sight of him, and not at the remembrance of his tears; for the last clause is to be read in a parenthesis, and these words stand not connected with that, but with the preceding part of the text. The apostle intimates, that a sight of his dearly beloved son Timothy would fill him with joy amidst all his troubles and afflictions he endured for the Gospel: this is an instance of hearty, sincere, and strong affection. ~John Gill

That I may be filled with joy — to be joined with “desiring to see thee” (Romans 1:11-12, Romans 15:32). ~Jamieson, Fausset, Brown

Quoted verses:
Romans 1:11-12
1 I say then, Hath God cast away his people? God forbid. For I also am an Israelite, of the seed of Abraham, of the tribe of Benjamin.
2 God hath not cast away his people which he foreknew. Wot ye not what the scripture saith of Elias? how he maketh intercession to God against Israel, saying,

Romans 15:32
That I may come unto you with joy by the will of God, and may with you be refreshed