[view·edit·purge]Mindflayers, or "Illithids," are a race found in Dungeons & Dragons. They appear humanoid, with heads that resemble octopuses, and frequently are found in the Underdark. They have fearsome psychic powers, subsist on a diet comprised of the brain tissue of sapient beings, and often enslave other races.
Like so many enemies in Final Fantasy's bestiary, Mind Flayers are taken directly from the bestiary of Dungeons & Dragons, and are one of the most infamous and iconic monsters of that game.
While the "Evil Eye" (AKA a Beholder) was drastically changed in appearance in every single release after the initial Japanese one due to copyright concerns, Mind Flayers have not been renamed or graphically altered, despite their prominence in Dungeons & Dragons, and have appeared in other games since. One general assumption is that, so long as the Mind Flayers are not called "illithids" (which is a specifically-protected term), Wizards of the Coast cannot claim copyright infringement, as the concept of the "octopus-faced mind eater" comes from still earlier works by Brian Lumley and H.P. Lovecraft.