Jamieson, Fausset and Brown on 1 Timothy 1:2

2. my own son--literally, "a genuine son" (compare Ac 16:1; 1Co 4:14-17). See Introduction.

mercy--added here, in addressing Timothy, to the ordinary salutation, "Grace unto you (Ro 1:7; 1Co 1:3, &c.), and peace." In Ga 6:16, "peace and mercy" occur. There are many similarities of style between the Epistle to the Galatians and the Pastoral Epistles (see Introduction); perhaps owing to his there, as here, having, as a leading object in writing, the correction of false teachers, especially as to the right and wrong use of the law (1Ti 1:9). If the earlier date be assigned to First Timothy, it will fall not long after, or before (according as the Epistle to the Galatians was written at Ephesus or at Corinth) the writing of the Epistle to the Galatians, which also would account for some similarity of style. "Mercy" is grace of a more tender kind, exercised towards the miserable, the experience of which in one's own case especially fits for the Gospel MINISTRY. Compare as to Paul himself (1Ti 1:14, 16; 1Co 7:25; 2Co 4:1; He 2:17) [Bengel]. He did not use "mercy" as to the churches, because "mercy" in all its fulness already existed towards them; but in the case of an individual minister, fresh measures of it were continually needed. "Grace" has reference to the sins of men; "mercy" to their misery. God extends His grace to men as they are guilty; His "mercy" to them as they are miserable [Trench].

Jesus Christ--The oldest manuscripts read the order, "Christ Jesus." In the Pastoral Epistles "Christ" is often put before "Jesus," to give prominence to the fact that the Messianic promises of the Old Testament, well known to Timothy (2Ti 3:15), were fulfilled in Jesus.