Jesus Christ (v.1) — the best texts have “Christ Jesus” both places in this verse. “Jesus Christ” describes Him in reference to His resurrection, as one who was rejected and then glorified. “Christ Jesus” emphasizes His preexistence. “Christ” is the Greek word for the Hebrew word “Messiah.” Both mean “the one who is anointed.”
In a Jewish setting such as the Gospel according to Matthew, the word [Christ] refers to the Messiah of Israel, the Anointed of God who is to become its King. In a Church setting, as here in First Timothy, it had the significance, not of the covenanted King of Israel, but of The Anointed One of God, to Paul and his Greek readers. The name “Jesus” is the English spelling of the Greek word Iesous, which in turn is the Greek spelling of the Hebrew word we know in its transliterated from as “Jehoshua,” the “h” disappearing, since the Greek language has no letter “h.” The Hebrew word means “Jehovah saves.” — Wuest, pages 22-23.
by the commandment (v.1) — Paul was under orders
God our Savior (v.1) — … That we may lead a quiet and peaceable life in all godliness and reverence. For this is good and acceptable in the sight of God our Savior, who desires all men to be saved and to come to the knowledge of the truth (1 Timothy 2:2-4. See also 4:10-11).
hope (v.1) — referring to that which is not yet seen — faith and assurance in the future
In the text there is an antithesis between the offices of God as our Savior and of Christ Jesus as our hope. The one points to the past, at least, chiefly, and the other to the future. In speaking of the saving action of God, Paul uses the aorist (2 Timothy 1:9; Titus 2:11; 3:4-5). He saved us potentially … God is the efficient cause of our justification, while Jesus, “our righteousness” besides being the meritorious cause, may be said to be the formal cause; for “the righteousness of God by which He maketh righteous,” is embodied in Jesus, who “was made unto us … righteousness and sanctification” (1 Corinthians 1:30). We advance from salvation to sanctification; and accordingly we must not narrow down the conception of Christ Jesus our hope to mean “the hope of Israel” (Acts 23:6; 28:20), but rather, the historical manifestation of the Son of God as Christ Jesus is the ground of our “hope of glory” (Colossians 1:27). Our hope is that “the body of our humiliation will be conformed to the body of His glory” (Philippians 3:20-21). Our hope is that we shall be like Him (1 John 3:2). — Wuest, pages 23-24.
Timothy (v.2) = he who honors God
true (v.2) = genuine, true-born, legitimately born
son (v.2) — Paul probably led Timothy to the Lord on his first visit to Lystra (Acts 14). It was likely that Timothy had been a saved Jew, according to Old Testament truth. Paul led him into the Body of Christ.
grace (v.2) = unmerited favor despite sin and its effects
mercy (v.2) = expression of pity — implies need in the one it’s given to — Paul only includes mercy in his salutation to Timothy, perhaps because of Timothy’s timidness. (It appears in Titus, but isn’t in the best manuscripts there.)
Grace is for the guilty, mercy for the miserable.
peace (v.2) = to bind together that which has been separated.
God the Father (v.2) — “God the Son” and “God the Holy Spirit” are never used in Scripture. The Bible refers to Them as “Son of God” and “Spirit of God.”
Wuest’s translation — Paul, an ambassador of Christ Jesus by command of God our Savior and Christ Jesus our hope, to Timothy, my genuine child in the Faith. Grace, mercy, peace, from God our Father and Christ Jesus our Lord. — Wuest, page 26.