There doesn't seem to be a definitive answer. However, there are a few possibilities...
According to the Christmas special "Santa Claus is Comin' to Town," her name is Jessica, and she's a schoolteacher in Sombertown. SantaClaus.com goes a step further, claiming her middle name is Mary. But are those "official" sources?
Other than the good-humored reply that her first name is "Mrs.," there's very little evidence or agreement on what her mama actually named her. In fact, this article from "Santa Land" acknowledges that nobody (not even Santa himself) seems to know her first name.
We do know that Mrs. Claus first appeared in 1890, in a book of poetry called "Sunshine and Other Verses for Children." The book's author, Katherine Lee Bates, also wrote the words to the song "America the Beautiful."
It took a song, performed by George Melachrino and his Orchestra in 1954 and Nat King Cole in 1956, to launch Mrs. Claus into popular consciousness. And considering Santa's workload, it's a good thing he has someone looking after him.