Would the average person in the Forgotten Realms call Mind Flayers "Mind Flayers" or "Illithid?"
The average person living on the surface almost certainly knows them as Mind Flayers, if they know them at all
Dragon #281 includes the article "By Any Other Name: Races of the Underdark", about the naming conventions of the various races that live in the Underdark, which explains that in their native telepathic communication mind flayers are identified by a sort of thought-image or pattern, but that:
... when members of the other races of the Underdark have need to refer to a particular illithid, they usually try to translate his identifying thought-image into the trade tongue of Undercommon.
This is accompanied by a couple of tables which suggest that the word "illithid" itself is formed from the two Undercommon words "illi", meaning "mind", and "thid", meaning "flayer/ruler" - so illithid is literally just the Undercommon word meaning "mind flayer".
Undercommon is a rare language on the surface, and mind flayers themselves are even rarer; it seems very unlikely that the term "illithid" would have made its way into other languages as a loanword, given the language would be very rarely encountered and the term very rarely useful. Undercommon speakers who had occasion to tell surface-dwellers about the illithid would presumably have just directly translated to "mind flayer" to preserve the meaning.