Jamieson, Fausset and Brown on 2 Timothy 1:2

2. my dearly beloved son--In 1Ti 1:2, and Tit 1:4, written at an earlier period than this Epistle, the expression used is in the Greek, "my genuine son." Alford sees in the change of expression an intimation of an altered tone as to Timothy, more of mere love, and less of confidence, as though Paul saw m him a want of firmness, whence arose the need of his stirring up afresh the faith and grace in Him (2Ti 1:6). But this seems to me not justified by the Greek word agapetos, which implies the attachment of reasoning and choice, on the ground of merit in the one "beloved," not of merely instinctive love. See Trench [Greek Synonyms of the New Testament].