There is a technical answer to this question, as well as a symbolic answer.
Technically, if you read the text closely, he probably died from some sort of feverish exhaustion. He spent so much time riding the horse, and obsessing about making money and being lucky for his mother, that he physically wore himself down. He didn't sleep or eat enough, and overexterted himself on the rocking horse. His little body couldn't take it anymore. Lawrence makes a point of describing his shiny eyes, his feverish glance, and his singlular focus. So, he probably technically died of overexertion and exhaustion.
Symbolically, it was greed, or materialism that killed little Paul. His mother constantly sought out more and more money, and in a conversation with Paul, revealed that luck is "what causes you to have money." Paul already sense that she "could not love [him]," so, in his mind, he thinks that if he can be lucky, and make money, then she will love him. His desire to be loved was spurred by her materialistic greed. Even when he does win money for her, it does no good. So, just as greed, and the desire for more money, is an insatiable and never-ending black hole of need, so was Paul's obsession with it, and his life was the sacrifice that, in the end, was paid.
I hope that those thoughts help a bit; good luck!
Although the answer above covers most of it, I would like to introduce another interesting outlook on this question. It could be argued that it was love that ultimately killed Paul. While on the surface it was riding the rocking horse which ultimately killed Paul, the reader must dissect Paul's motivations to find the real reason behind his death. It is Paul's desire to be loved that ultimately kills him. However, if we go a layer deeper the theory arises that to love, you must love first. It can be argued that Paul won so much money and rode the rocking horse because he loved his mother and it was this love that ultimately killed him.