We are, of course, told almost nothing about the hopes of the prodigal son. Therefore, we have to use our own understanding of human nature to try to understand what his motivation was.
There are two main things that the prodigal son might have hoped to gain by leaving home. First, we will assume that the prodigal son was rather young and perhaps still an adolescent. His father was rich and successful. His older brother, it would appear, always acted in the right ways. The son might well have felt overshadowed by his father and his older brother. He might have wanted to make his own way in the world so that he could feel important in his own right. By doing so, he could hope to gain more of a sense of self-worth. Second, the prodigal son might have felt that there was going to be no place for him at home in years to come. As the second son, he might have felt that his older brother was going to be the head of the family and that he, himself, might remain in a subordinate position all his life. If this is the case, he could have hoped to gain some independence by going off to live on his own.