The rocking-horse seems to be a symbol of the futility of Paul's quest to win his mother's love. She is incapable of loving her children, according to the narrator. In fact, this woman is incapable of loving anybody.
She had bonny children, yet she felt they had been thrust upon her, and she could not love them....Only she herself knew that at the centre of her heart was a hard little place that could not feel love, no, not for anybody.
Paul believes that if he can become lucky and win lots of money his mother will love him. But she is incapable of loving anybody. No matter how hard he rides his rocking-horse and how furiously he lashes it, in the end the rocking-horse is still standing in exactly the same place. The unfortunate boy can never get what he really wants, regardless of how much money he wins. The rocking-horse symbolizes Paul's natural desire for his mother's love, his strong motivation to gain it, his anxiety and desperation, and the utter hopelessness of his childish dream. The rocking-horse also symbolizes Paul's helplessness. He is just a little boy. What other possible means would he have of making a large amount of money to please his mother? None.
I would definitely want to argue that it is. Throughout this excellent short story it is the rocking horse that gives Paul his supernatural ability to predict the next big winner, and, arguably, it is the rocking horse that takes his life as he rocks ever more frantically on the horse to gain this knowledge. Note how the mother views her son on his horse for the last time:
Then suddenly she switched on the light, and saw her son, in his green pajamas, madly surging on the rocking horse. The blaze of light suddenly lit him up, as he urged the wooden horse...
He screams out "in a powerful, strange voice" and his eyes "blaze" with the effect of the horse. It is clear that the horse does symbolise some sort of power external to Paul and his mother, yet we are never given a precise indication as to what. We can, however, come up with a list of various possibilities. We could say that the horse represents Paul's desire for his mother's love, an instrument of supernatural forces, temptation or greed. Any of these possibilities could be argued from the text.