Here is what Memory Alpha has to say about her name. The upshot is there has never been a definitive canonical name assigned to her character, but she has been given a long string of possible names in various apocryphal works.
In the Marvel Star Trek: Early Voyages comic book series, this character was named Lieutenant Commander Robbins. Her first name was interrupted by other dialogue, but starts with "Eure—."
The novel Vulcan's Glory, by Star Trek: The Original Series writer D.C. Fontana, suggested this character's moniker was not simply a nickname or title — she was an Illyrian who was called "Number One," as the best intellect among her generation. The novels The Children of Kings and Child of Two Worlds explain that Number One does have a given name, but it's difficult to pronounce for non-Illyrians; the former suggests that she liked to be called by this name rather than her own as it enabled her to maintain a professional relationship with her captain. In the novel Captain to Captain she is referred to as "Una", however this is not her real name either.
In the Captain's Table novel Where Sea Meets Sky, Number One was given the last name "Lefler" (an allusion to Robin Lefler).
Star Trek II: Biographies gives her real name as Leigh Hudec.
Peter David's New Frontier series of novels had a character named "Morgan Primus" who was an immortal like Flint (from TOS: "Requiem for Methuselah") and was the mother of Robin Lefler. It is hinted that she and Number One are the same person.
In the Star Trek: Discovery novel, Desperate Hours, Number One had been promoted to Commander and was given the last name "Una".