Rather than “greed,” I would say the theme of Lawrence’s story was “luck,” or rather, the lack of it. This is the mother’s problem: despite her seeming affluence, she is “unlucky”—there is a “hard little place” at the center of her heart that keeps her from feeling love for her children—a secret only she and her children know.
The rest of the story can be read as an attempt to explain what “luck” might be. It’s clear as the story progresses that genuine parental love, and the idea of a nurturing home life, have been replaced by a house that constantly whispers “there must be more money.” The rocking horse is not a plaything but a way of “being sure”—a tool of Paul’s “trade” of being lucky. Although the mother tells Paul that being lucky is better than being rich, by the end of the story the eighty thousand pounds Paul makes at the Derby is not enough to get around the hard place in his Mother’s heart. She has, unwittingly, traded her son for a lot of money. As her brother says to her at the end of the story, “’My God, Hester, you're eighty-odd thousand to the good, and a poor devil of a son to the bad. But, poor devil, poor devil, he's best gone out of a life where he rides his rocking-horse to find a winner.’” The implication is that it is only in death that Paul is truly lucky.
Greed is a major theme of this story. Paul is a selfless boy wanting to help his family. His mother is a greedy and selfish woman, wanting not to make the family comfortable, but to buy pretty things. Her greed causes her to lie and sneak, hiding what should be shared from her husband and the rest of the family. Lawrence not only demonstrates the negative effects of her greed - Paul's death and the conflict within the family - but he also demonstrates the effect of greed on itself. Greed increases unto itself. For a greedy person, the more they get, the more the want. The are never satisfied, as is clearly true of the mother in this story. She could have $1000 a year to make her comfortable, but she demands all the money at once - which of course leads her to spend it all at once. Ultimately, her greed destroys Paul and, ironically, the chance the family had to become undeniably wealthy.