Why does Paul ride his rocking horse?

This answer will change depending on when in the story readers find Paul riding his horse; however, the two parts of the story are very close together. The first time that readers see Paul riding the horse, he is riding the rocking horse because it is a cathartic activity. It helps him clear his head and think about the problem of finding luck for his mother.

He went off by himself, vaguely, in a childish way, seeking for the clue to "luck." Absorbed, taking no heed of other people, he went about with a sort of stealth, seeking inwardly for luck. He wanted luck, he wanted it, he wanted it. When the two girls were playing dolls in the nursery, he would sit on his big rocking-horse, charging madly into space, with a frenzy that made the little girls peer at him uneasily. Wildly the horse careered, the waving dark hair of the boy tossed, his eyes had a strange glare in them. The little girls dared not speak to him.

Later, Paul rides the rocking horse for a different reason. The horse does indeed bring him luck. When Paul rides the rocking horse, he is able to predict the winning horse in upcoming horse races.

Paul's secret of secrets was his wooden horse, that which had no name.

Paul, Oscar, and Bassett use this "lucky" skill of Paul's to win money by betting on horse races. Paul rides the rocking horse to make money in order to help make his mom less depressed by their financial situation.