Mind Flayers seem to look a whole lot like Cthulhu. Is there any relationship/inspiration between the two?
Indirectly, yes. Gary Gygax tells us that "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." Said novel was firmly rooted in the Cthulhu mythos; in fact, Lovecraft's character Robert Harrison Blake wrote a short story titled "The Burrower Beneath." We may suspect that title was an influence on Brian Lumley's choice of titles.
The cover in question, for reference:
As Graham points out the illithids share more physical similarity to Chthonians.
Chthonians (/ˈθoʊniənz/ from Greek: chthon, "earth") are fictional creatures in the Cthulhu Mythos. Chthonians as name of the species is the creation of English horror-fiction writer Brian Lumley and was first featured in his short story "Cement Surroundings" (1969)—though the creature never made a direct appearance. The chthonians had a more prominent role in Lumley's novel The Burrowers Beneath (1974), whose title was taken from one of the stories said to have been written by Robert Blake in Lovecraft's "The Haunter of the Dark."
The brain sucking probably refers to "The Space-Eaters" by Frank Belknap Long.
I think they were definitely inspired by Cthulhu, and there's no secret that the original D&D designers and fans were very familiar with the works of HP Lovecraft. It's probably not just illithids, either. We could look closely and see a lot Mythos-inspired monsters in the earliest monster manuals as well as the newest ones.
The Cthulhu Mythos was featured in the original print-runs of Deities and Demigods.