As you suggest, the number "3" is significant in the short story "The Monkey's Paw" by W.W. Jacobs.
The structure of the story is full of threes. For example, it is written in three parts. All three parts take place at three different times of the day, and we cannot ignore the three different types of weather in each section: rainy, sunny, and windy cold.
The most obvious and memorable use of 3 is the three wishes, traditional in all "wish" stories such as genies and fairy godmothers. Even more, this talisman will grant three wishes to three different holders of the monkey's paw--and Mr. White is the third owner.
Sergeant-Major Morris comes to visit the Whites, a family of three, and he has three glasses of whiskey before he shares his intriguing adventures in India. When White indicates his interest in the monkey's paw, the sergeant-major tries three times to convince White that he does not really want the token.
When he has to deliver the bad news about Herbert's death, the man from Maw and Meggins approaches the Whites' gate three times before he finally feels strong enough to deliver the blow. Later, when White considers retrieving the paw from the fire, his wife implores him three times to wish that their son were alive again. After he does, Herbert can be heard knocking at the door three times before Mrs. White finally hears him.
Why did Jacobs choose 3? We can only speculate, but consider the following ideas. In the natural world, nearly everything is in pairs (twos). Consider the human body, for example, which of course is comprised of many pairs: eyes, ears, limbs, lungs. The suggestion of three, at least in a literary sense, is the suggestion of something grotesque and unnatural.
The old adage that bad luck always comes in threes is no doubt at least part of the premise for the author's deliberate us of three throughout the story. Another obvious "three" which may play a role in this story is the Holy Trinity: the Father, the Son, and the Holy Spirit. This story seems to be a combination of these two ideas. By ignoring the spiritual realm and giving credence to a distinctly non-Christian talisman in the form of a monkey's paw, the Whites have brought their troubles on themselves.
For mor insights and analysis of this and other classic short story, visit the excellent eNotes sites linked below.