What makes Bud so self-conscious about his name?
The most important thing to Bud is his name. He was told by his mother to never let anyone call his Buddy, so he is self-conscious about people calling him Buddy. His mother named him Bud and now that she is dead he isn't going to let anyone call him otherwise. Bud's mother gave him his name for a special reason.
Bud insists that his name is an adult name and not that of a child. Bud has had to grow up quickly since his mother died. He spends time running from the impersonal confines of a 1930s orphanage to dysfunctional foster homes. Bud searches for his birth father. The only clue he has to his father is a blue flyer advertising Herman E. Calloway and the Dusky Devastators of the Depression, a jazz band working outside of Grand Rapids, Michigan.
Bud constantly corrects people when they call him Buddy. Bud is indignant when the "pretend" mission family calls him Clarence until he figures out the game and even then he thinks there might have been a better one. In the end, Bud receives a new name from the jazz band musicians and is initiated into the group. His identity is secure.