Seagulls are believed to carry the souls of lost sailors.1 Therefor, killing one is supposed to bring bad luck (i.c. a storm).
Pattinson's Ephraim shares similarities with the figure of Prometheus, climbing mount Olympus to steal fire (as in divine wisdom, guarded by Dafoe's Thomas, a character akin to Proteus), and being eaten alive by a bird as punishment.2 Maybe the one-eyed seagull is a portent of the role Ephraim will play, and his demise.
These two theories do not account for the monocular quality of the bird, though.
A very practical explanation would be that a one-eyed seagull is easy to discern from their relatives, communicating to the audience that we deal with an individual gull, and not just different ones.
Perhaps the seagull represents the Graeae, the three sisters from Greek mythology who shared one eye, and represent dread, alarm, and horror. This would play well with the fact that 'one-eye' in the film was played by three actual seagulls, Johnny, Lady, and Tramp.3 But this is pure speculation.
Ultimately, we don't know. Director Robert Eggers has been evasive in answering questions of this kind, simply because he wants the audience to wonder and make up their own minds:
"Look, some movies, you want to know everything, and it’s very clear. This isn’t that kind of movie. And I’ve read some incredibly well-done dissections of Lynch’s Lost Highway, and it’s like I don’t want to read that—I don’t want to know all the answers. I like having to wonder. That’s what’s fun about a movie like this." 4
That is not to say the brothers Eggers (Robert co-wrote The Lighthouse with his brother Max) don't have answers for themselves, however: "Me and my brother had to have answers to all these questions to write this thing."