Paul's mother defines luck as that which "causes you to have money." She continues explaining to Paul:
If you're lucky you have money. That's why it's better to be born lucky than rich. If you're rich, you may lose your money. But if you're lucky, you will always get more money.
Paul's mother was attractive, married for love, and had beautiful children, but something was missing from her life. She always felt "the centre of her heart go hard" when her children were present. This is because she wanted more money. She associated her potential happiness with money. Therefore, she describes luck (to Paul) as that which causes one to have money, that which causes one to become happier.
Paul, going strictly by this definition, determines that luck is some mysterious psychological force that is only associated with acquiring money. From his discussions with Bassett, he determines that it would make the most sense to use luck to gamble and thereby acquire more money. The acquisition of money becomes all-encompassing. Paul just wants to quiet the whisper of the house, "There must be more money!" It is this constant emphasis on money, from his mother (but partially enabled by Uncle Oscar and Bassett) that conditions Paul to equate using luck to find money as the family's greatest necessity. It is a lost cause because Paul's mother (Hester), obsessed with money, is also irresponsible with it. She is wearing an expensive white fur coat (after receiving the money from Paul's winnings) when she goes to check on Paul near the end of the story. This is money she should have used to pay debts or for the children's schooling.