The human weakness that is displayed by the White family demonstrates perfectly the point of the monkey's paw, and what it was meant to show humans. Note what the sergeant-major tells the White family about the monkey's paw, and the reason why it was made:
"It had a spell put on it by an old fakir," said the sergeant-major, "a very holy man. He wanted to show that fate ruled people's lives, and that those who interfered with it did so to their sorrow. He put a spell on it so that three separate men could each have three wishes from it."
The fakir created the monkey's paw to show that it was foolish to try and interfere in the fate that governs our lives as humans. The Whites, however, fall into just this trap, foolishly wishing for a sum of money, and then going on to make a wish for their son to return to them before finally realising the error of their ways and wishing that he would disappear again. The weakness of the Whites is therefore shown to be the desire to try and fight against their fate and change it, not being happy with what they are given by fate, and wanting more. The disastrous results show the truth of the fakir's objective: it is foolhardy and dangerous in the extreme to try and interfere in your own fate.