The horror cliche about zombies eating brains doesn't actually date back to zombie godfather George A. Romero. While hardcore fans of the zombie genre likely know their history, more casual movie watchers will likely picture some brain-eating when zombies are brought up. It's easy to see why as somewhere along the way, pop culture at large seemed to absorb the idea that zombies eat brains. Even The Simpsons, a pop culture benchmark if there ever was one has made jokes about brain-eating zombies.
What's strange is that for how often the concept of zombies eating brains is brought up or referenced, it's really not commonly used. In the pioneering zombie movies directed by George Romero, zombies eat human flesh, for seemingly no other reason than a primitive drive to do so. This is also true of AMC's TV juggernaut The Walking Dead, and most other notable zombie fiction. While zombies might end up eating some brain while devouring someone's face, that's more by happenstance than intent.
Romero essentially created the zombie genre as it's known today with 1968's Night of the Living Dead, albeit without ever actually referring to his creatures as such and it's telling the movie was almost titled Night of the Flesh Eaters. Eating flesh is just that big a part of the genre. The concept of zombies eating brains, in particular, was invented by cult classic 1985 horror comedy The Return of the Living Dead, written and directed by Dan O'Bannon, who also co-wrote Ridley Scott's original Alien movie.
According to all available evidence, O'Bannon's script for The Return of the Living Dead was the first pop culture instance of zombies eating brains. Interestingly, one of the zombies even explains why in the film, as Return of the Living Dead's zombies still retain some of their intelligence. The zombie says that, for reasons unknown, eating the brains of the living eases the "pain" of being dead. So in this case, the zombies aren't eating brains out of hunger, it's to satiate the relentless pain that apparently comes with rising from the grave. Somewhere along the way after the film's release, this became part of zombie lore.
The Return of the Living Dead connects to Romero in some small ways, as its existence owes to a creative parting of ways by Romero and Night of the Living Dead producer John A. Russo. Return also reveals a version of the events of Night actually happened in-universe. Still, Romero had no part in Return of the Living Dead's creation but ended up having some encounters with fans over the years in which they'd want him to write something about eating brains on a signed item, leaving him bemused due to his zombies never actually doing that.