This is a very personal letter. From elsewhere in the New Testament we know that Paul was very fond of Timothy; he speaks of his love for the younger man and of his conviction that he was faithful (1 Cor. 4:17). Paul says further that Timothy could remind the Corinthians of Paul’s way of life, which indicates a certain intimacy and shows that Paul trusted him. It accords with this that he likens Timothy’s relationship to him to that of a son to his father (Phil. 2:22), and with a cheerful disregard for consistency speaks of him as a brother (and fellow worker, 1 Thess. 3:2). He links Timothy with himself in the opening greetings in some of his epistles (2 Cor. 1:1; Phil. 1:1; Col. 1:1; 1 Thess. 1:1; 2 Thess. 1:1), which argues that he was a trusted colleague. Paul asks the Corinthians to ensure that Timothy “has nothing to fear” if he should visit them (1 Cor. 16:10), which seems to indicate a certain diffidence about the young man. He sent him to the Thessalonians, he assures them, “to strengthen and encourage you in your faith” (1 Thess. 3:2), and he plans to send him to the Philippians, explaining, “I have no one else like him, who takes a genuine interest in your welfare” (Phil. 2:20).
All this gives point to Paul’s greeting, “To Timothy my true son in the faith” (1 Tim.1:2). The letter is written to a younger man for whom the apostle had a deep affection and whom he had for years entrusted with important missions. What Paul now says brings out the truth that Christians are linked in the service of the Lord and that there is significant help they can and should give to one another. - D. A. Carson & Douglas J. Moo - An Introduction to the New Testament