From the perspective of Paul's mother, Hester, the main problem with the family is that they are unlucky. The narrator, using the mother's point of view, states:
There was a beautiful woman who started life with all the advantages, but she had no luck.
We learn that she married for love, but the love crumbled into dust. We also learn that she doesn't love her three children.
To the mother, the family is "poor" because they have no luck. She says to Paul, when he asks why they are "poor:"
it's because your father has no luck.
When Paul asks if luck is the same thing as money, she says, no, but it is what brings money to a household.
Of course, it is only the mother's perception that the family is poor. From all the descriptions we have of their lifestyle, they live very well. They own a fine house with a garden, employ servants, and the children have a nursery full of toys.
The mother feels poor because she is incapable of loving her husband or children. This leaves her with a hole inside that she tries to fill up with material goods. But as the story shows, no amount of money or goods is enough to compensate her for her inner emptiness.
Hester externalizes her inner emptiness by blaming her sense of lack on bad luck. Unfortunately, her missing piece impacts the whole family. The very walls of the house seem to be saying that the family needs more money, while Paul rocks himself to death to try to get his mother the money she needs and so earn her love.