The mind flayer I made up out of whole cloth using my imagination, but inspired by the cover of Brian Lumley's novel in paperback edition, The Burrowers Beneath.Now, I've generally come to think that, while these Q&A threads are quite useful, the answers given must sometimes be taken with a grain of salt, if only they were often asking for information that's more than 30 years old. I don't know about you, but I often find it very difficult to present my recollections of several decades in the past in a totally unbiased fashion. I don't think Gary was any different.
In this case, though, I am inclined to accept Gygax's claim at face value. The specific details he provides in the quote above suggest an accurate memory. Brian Lumley's Lovecraft-inspired novel was first released in paperback in the United States in early 1974, about a year before the mind flayer first appeared, so the timeframe is right. I think it quite plausible that Gary had a copy of the book in his home and was inspired by its cover artwork to create a race of subterranean tentacular beasties as new adversaries in D&D.