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 other commentaries

Our verse again is verse 3.

2 Timothy 1:3
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;

Now to the other commentaries. First the general and then the specific.

The Matthew Henry Concise speaks to 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

I would take this concept a bit further to include all of the interventions by God directly and indirectly in the lives of future firstfruits from birth. Stories could be told on how God used many methods of putting ideas into the heads of individuals He would later call. He leads them to information, His Word and to concepts and principles. Lessons are taught, such as those about not lying. Principles are passed to them via teachers, parents, strangers and employers. Some come by early television which taught moral or even spiritual concepts. Clearly God is working with future firstfruits from birth.

Now to the specific commentaries. In my view of the commentaries I noticed that they differed on how they broke out the verse into its parts so rather than dealing with this verse by its parts, we will take one commentary at a time. First, the Barnes Notes.

I thank God, whom I serve from my forefathers - Paul reckoned among his forefathers the patriarchs and the holy men of former times, as being of the same nation with himself, though it may be that he also included his more immediate ancestors, who, for anything known to the contrary, may have been distinguished examples of piety. His own parents, it is certain, took care that he should be trained up in the ways of religion; compare the Philippians 3:4-5; Acts 26:4-5.

Quoted verses:
Philippians 3:4-5
4 Though I might also have confidence in the flesh. If any other man thinketh that he hath whereof he might trust in the flesh, I more:
5 Circumcised the eighth day, of the stock of Israel, of the tribe of Benjamin, an Hebrew of the Hebrews; as touching the law, a Pharisee;

Acts 26:4-5
4 My manner of life from my youth, which was at the first among mine own nation at Jerusalem, know all the Jews;
5 Which knew me from the beginning, if they would testify, that after the most straitest sect of our religion I lived a Pharisee.

The phrase “from my forefathers,” probably means, after the example of my ancestors. He worshipped the same God; he held substantially the same truths; he had the same hope of the resurrection and of immortality; he trusted to the same Saviour having come, on whom they relied as about to come. His was not, therefore, a different religion from theirs; it was the same religion carried out and perfected. The religion of the Old Testament and the New is essentially the same; see the notes at Acts 23:6.

Quoted verse:
Acts 23:6
But when Paul perceived that the one part were Sadducees, and the other Pharisees, he cried out in the council, Men and brethren, I am a Pharisee, the son of a Pharisee: of the hope and resurrection of the dead I am called in question.

With pure conscience - see the notes at Acts 23:1.

Quoted verse:
Acts 23:1
And Paul, earnestly beholding the council, said, Men and brethren, I have lived in all good conscience before God until this day.

Notice the commentary on Acts 23:1:
 

I have lived in all good conscience - I have conducted myself so as to maintain a good conscience. I have done what I believed to be right. This was a bold declaration, after the tumult, and charges, and accusations of the previous day Acts 22; and yet it was strictly true. His persecutions of the Christians had been conducted conscientiously, Acts 26:9, “I verily thought with myself,” says he, “that I ought to do many things contrary to the name of Jesus of Nazareth.” Of his conscientiousness and fidelity in their service they could bear witness. Of his conscientiousness since, he could make a similar declaration. He doubtless meant to say that as he had been conscientious in persecution, so he had been in his conversion and in his subsequent course. And as they knew that his former life had been with a good conscience, they ought to presume that he had maintained the same character still. This was a remarkably bold appeal to be made by an accused man, and it shows the strong consciousness which Paul had of his innocence. What would have been the drift of his discourse in proving this we can only Conjecture. ~Barnes Notes

That without ceasing - compare Romans 12:12; 1 Thessalonians 5:17.

Quoted verses:
Romans 12:12
Rejoicing in hope; patient in tribulation; continuing instant in prayer;

1 Thessalonians 5:17
Pray without ceasing.

I have remembrance of thee in my prayers night and day - see the notes at Philippians 1:3-4. ~Barnes Notes

Quoted verse:
Philippians 1:3-4
3 I thank my God upon every remembrance of you,
4 Always in every prayer of mine for you all making request with joy,

Now to the Adam Clarke:

Whom I serve from my forefathers - Being born a Jew, I was carefully educated in the knowledge of the true God, and the proper manner of worshipping him.

With pure conscience - Ever aiming to please him, even in the time when through ignorance I persecuted the Church.

Without ceasing I have remembrance of thee - The apostle thanks God that he has constant remembrance of Timothy in his prayers. It is a very rare thing now in the Christian Church, that a man particularly thanks God that he is enabled to pray for Others. And yet he that can do this most must have an increase of that brotherly love which the second greatest commandment of God requires: Thou shalt love thy neighbor as thyself. It is also a great blessing to be able to maintain the spirit of a pure friendship, especially through a considerable lapse of time and absence. He that can do so may well thank God that he is saved from that fickleness and unsteadiness of mind which are the bane of friendships, and the reproach of many once warm-hearted friends. ~Adam Clarke

Now to the John Gill:

I thank God - After the inscription and salutation follows the preface to the epistle; which contains a thanksgiving to God upon Timothy's account, and has a tendency to engage his attention to what he was about to write to him in the body of the epistle. God is the object of praise and thanksgiving, both as the God of nature and providence, and as the God of all grace; for every good thing comes from him, and therefore he ought to have the glory of it; nor should any glory, as though they had not received it: and he is here described, as follows,

whom I serve from my forefathers with pure conscience - the apostle served God in the precepts of the law, as in the hands of Christ, and as written upon his heart by the Spirit of God, in which he delighted after the inward man, and which he served with his regenerated mind; and also in the preaching of the Gospel of Christ, in which he was very diligent and laborious, faithful and successful: and this God, whom he served, was the God of his "forefathers", of Abraham, Isaac, and Jacob, and of Benjamin, of whose tribe he was, and also of his more immediate ancestors. The Ethiopic version renders it, "from my original"; for though he preached the Gospel of Christ, and asserted the abrogation of the ceremonial law, yet he worshipped the one, true, and living God, the God of Israel, and was not an apostate from the true religion, as his enemies would insinuate: and this service of his was performed with a "pure conscience": every man has a conscience, but the conscience of every natural man is defiled with sin; and that is only a pure one, which is sprinkled and purged with the blood of Christ; and whereby a person is only fitted to serve the living God, without the encumbrance of dead works, and slavish fear, and with faith and cheerfulness; and such a conscience the apostle had, and with such an one he served God. For this refers not to his serving of God, and to his conscience, while a Pharisee and a persecutor; for however moral was his conduct and conversation then, and with what sincerity and uprightness soever he behaved, his conscience was not a pure one. He goes on to observe what he thanked God for,

that without ceasing I have remembrance of thee in my prayers night and day - that God had laid him upon his heart, and that he had such reason to remember him at the throne of grace continually. We learn from hence, that the apostle prayed constantly night and day; and if so great a man as he stood in need of continual prayer, much more we; and that in his prayers he was not unmindful of his friends, though at a distance from him; and in both these he is to be imitated: it becomes us to pray without ceasing: to pray always, and not faint and give out, to pray every day and night; and to pray for others as well as for ourselves, for all the saints, yea, for our enemies, as well as for our friends. ~John Gill

Recap
1] The leading subject of ministers is Salvation -The Salvation Process - Eternal Life in the Kingdom of God.
2] True ministers are in close adherence to the Word of God.  They preach from the Word of God.
3] The faith of firstfruits is unfeigned; it will stand the trial.
4] Faith dwells in firstfruits as a living principle.
5] Firstfruits today hold the same religion and the same doctrines and beliefs as those at the beginning of the New Testament church.
6] Firstfruits conduct themselves in the Salvation with a pure conscience.  They are continually moving forward in the Salvation Process and addressing the elements of it.
7] Firstfruit are ever aiming to please God.  This is part of having a pure conscience.
8] Firstfruits are patient in tribulation and continue instant in prayer.
9] We give thanks to God continually and put everything into His hands [Assiduity].
10] We pray for the brethren continually.