Although his wish is not specified, he seems to have wished that whatever was causing the knocking at the door would go away. Because the second wish was that his son be alive again, his wife believes that the son is knocking at the door. Too late Mr. White realizes that if indeed his no-longer-dead son has returned, he will appear mangled and bloody because of the horrible death he suffered; Mr. White didn't have the foresight to wish him alive again whole and healthy. Therefore, he uses his final wish to make his son disappear, and his experience with the monkey's paw fulfills what Sgt. Major Morris had predicted: those who interfere with fate do so to their sorrow.
The author takes pains to establish that only Mr. White can make wishes with the monkey's paw. The Sergeant-Major tells the family:
"He put a spell on it so that three separate men could each have three wishes from it."
W. W. Jacobs inserts some additional dialogue to establish that Mr. White is the sole, legitimate owner of the monkey's paw.
"Did you give him anything for it, father?" inquired Mrs. White, regarding her husband closely.
"A trifle," said he, colouring slightly. "He didn't want it, but I made him take it. And he pressed me again to throw it away."
This would prevent Mrs. White from making any of the wishes. Otherwise she would have made the second wish herself. She has to insist on her husband's wishing for their son to come back to life and return home. Then it is entirely Mr. White's decision to make the third wish. He wishes for the second wish to be undone. He knows that his son must look like a horrible monster and doesn't want to have to look at him or have anything further to do with him. The thought of having such a creature living under their roof is unbearable. Mrs. White is terribly disappointed when she finally gets the door open, but the reader feels vastly relieved that there is no one outside.
The First Wish: For 200 Pounds [this leads to Herbert's death]
The Second Wish: For Herbert to come back to life [Herbert's ghost knocks on the door]
The Last Wish: For Herbert to rest in peace [this causes the ghost to go away and the knocking to stop].