On page 46 of his autobiography, My Life, Bill Clinton says that he's "still not sure exactly" why he decided, as a teenager, to change his name from Blythe, his biological father's surname, to Clinton, his stepfather's surname. He does, however, offer several possible reasons.
Firstly, his younger brother, already a Clinton, was about to start school, and Bill "didn't want the differences in our lineage ever to be an issue for him."
Secondly, he "just wanted the same name as the rest of (his) family." He had been using the name Clinton for several years anyway, so legally changing his name didn't seem like a very big step. And Bill's stepfather, Roger Clinton, had been in his life for almost a decade by this time, having married Bill's mother in 1950. Bill's biological father died in 1946, three months before Bill was born.
Thirdly, he speculates that he changed his name possibly just "to do something nice" for his stepfather. This was at a time when Bill's mother had decided to remarry Bill's stepfather, who she had recently divorced because he was violent. Bill expressed doubts about his mother's decision to take Roger back, but "understood her feelings." It is implicit in this part of the book that Bill changed his name as a gesture of solidarity with his mother and as a sign of his trust in her decision.