I like the idea that her real name is never officially given, so it's a moot point.I like what IDW comics have done with her. Number One eventually went on to command her own Constitution-class starship and was a commodore during TOS and an admiral by the time of TMP.
After leaving the Enterprise, she was no longer addressed as Number One, but just by her rank at all times (we never did learn her real name).
Read the New Frontier novels. Her name is revealed there.
Yeah, yeah, I know, novels aren't canon, but I don't care.....