Money is only important in the story because Paul's parents make it a priority, and establish it as the most important thing. We learn immediately that his mother is a bit fixated on money, and on maintaining a certain lifestyle. The parents like to live well, to keep up appearances, and money is needed for all of that. The first related conversation with her son, Paul, is about money, and about how she longs for more of it. She equates it to luck, and says that his father does not have luck. Paul, sensing her unhappiness, and desiring to feel love from her in any form, is able to recogonize even at that young age that his mother loves money most in her life. So, he sets out to get her money.
So, it is only because the characters themselves value money so much that money is important in the story. The author is using the unfortunate and tragic story as a warning to all, trying to make the point that money really isn't important at all; the love of money will only lead to sadness and loss. Greed is a hungry monster that is never fed; it only takes and takes until nothing is left. That is the point that Lawrence makes in this story, and he used characters that exemplify greed and the bottomless pit of money-hunger well. That love of money is symbolized in the personification of the house whispering, "There must be more money!" which haunts everyone there, driving Paul further and further into his obsession. Lawrence makes greed a living, breathing, ominou presence in the house, which it truly is if you think about it.
I hope that those thought helped; good luck!