In Bud, not Buddy, how does Bud change over the course of the story?
You have put your finger on a major theme of this story, which, after all, is a kind of coming of age story where we are presented with a narrator who, as an orphan, is left to himself to make his way in the world and find who he is. What is interesting is that he starts off the novel as a very street-wise orphan who has learned hard lessons from life already, in spite of his young age. Note the irony in Bud's repetition of "Here we go again" when he is told about a new foster home - Bud's fourth, and his prediction of the bullying he is going to receive from Todd Amos. It is clear that in a sense we are presented with a world weary and wise character who is old before his years.
Yet as the novel develops, the wisdom that Bud has gleaned from his experiences which he repeatedly quotes to us in the form of "Bud Caldwell's Rules and Things for Having a Funner Life and Making a Better Liar Out of Yourself." Yet at times the narrative shows how wrong his "Rules" actually are as they are based on distrusting others and find it hard to accept the kindness and charity of other characters in the novel, like his "pretend family" at the Mission who ensure he gets his breakfast and share their sugar with him, and also Lefty Lewis. Finally he finds his home and a family in the form of his grandfather and his band - and also he regains the ability to cry again.
It is important to remember the significance of Bud's name - as he remembers his mother explaining her reasoning for calling him "Bud" and not "Buddy":
"A bud is a flower-to-be. A flower-in-waiting. Waiting for just the right warmth and care to open up. It's a little fist of love waiting to unfold and be seen by the world. And that's you."
What is clear is that by the end of the novel Bud has found the place he needs to fulfil the promise in his name - he has found his home, giving him the security and love he needs to unfurl.