In The Absolutely True Diary of a Part-Time Indian, what makes Junior strong enough to leave when nobody else seems to have the courage?
What gives Junior the courage to leave the reservation more than anything else is the conversation he has with Mr P after throwing a book at him where Mr P tells him in no uncertain terms that he has to leave the reservation, saying that if he doesn't, he is going to be "killed" by everybody else eventually, including himself. Mr P doesn't literally mean killing, but he does mean that the unique person that Junior is who has so much potential is going to vanish if he stays on the reservation for a long time, as he will become just like everybody else. Junior is different, as the following quote describes:
You've been fighting since you were born... You fought off that brain surgery. You fought off those seizures. You fought off all the drunks and drug addicts. You kept your hope. And now, you have to take your hope and go somewhere where other people have hope.
It is these words above all else that show Junior the truth of what Mr P is saying. He is a character who, unlike the rest of his friends and family, has not lost hope. Because of this, he deserves and is capable of achieving a better future. After this conversation, Junior sets himself on a course to leave the reservation and find more and more hope as he goes on through life.