Take, for example, this comic imagining Zoro’s training with Mihawk by Twitter user @mugibaras. In it, Zoro, while anxious to advance after showing progress in his training, is quickly humbled by his teacher, resulting in the loss of his eye.
It’s entirely possible Mihawk noticed through their duel and training that Zoro suffers from ocular dominance in his left eye. That is to say, the left eye is stronger than the right. As a result, maybe Mihawk scarred the left eye shut and instructed Zoro not to open it until his right eye was trained enough.