I've always been a little unconvinced about this particular story. Memory Alpha seems to suggest "Kim Noonian Wang": https://memory-alpha.fandom.com/wiki/Khan_Noonien_Singh#Background_information
But, I dunno... it seems like a little bit of an odd way to reach out to an old friend. This friend was supposed to be Chinese... if so, why give him the additional decidedly un-Chinese names of Khan and Singh (and in more prominent positions)?
There's also an odd pan-Asian quality to the name: Khan is a South Asian name, usually given to Muslim men; Noonien is apparently supposed to be Chinese; and Singh is usually associated with members of the Sikh faith.
This calls to mind the origin of Sulu's name: the Sulu Sea, apparently so named because Roddenberry wanted Sulu to represent all of Asia.
In both cases, we have this peculiar situation of various Asian backgrounds being smooshed together into a single character. Now, don't get me wrong, there are lots of people with mixed Asian backgrounds, so it's not that either Sulu or Khan are impossible. But... I dunno. It just makes me think that Roddenberry came up with a cool-sounding name and then figured out an impossible-to-prove backstory to provide for it.