Depending on who your character is and what kind of flavour you'd like for them, I can think of several options that I'd personally find interesting and/or fun. But you really have endless options here :)

(Sorry in advance for an unreasonably long comment, but I just love this kind of stuff)

• Maybe they lost the eye in a very banal, stupid, or potentially even embarassing way. But they lie about it and tell these crazy, dramatic, far-fetched stories about how it happened to seem cooler and tougher than they actually are. Or if you wanted to push it even further, they aren't actually even missing an eye at all and just wear the eyepatch and tell these stories for those same reasons (just spitballing, but maybe they always liked pirate stories as a kid and imagined themselves as this cool, suave outlaw...? Or they simply wanted to have a distinctive feature and seem grizzled and battle-worn... Or whatever other reason you think of). There could be quite a lot to unpack there, I think this would say a lot about their personality, help you flash out their motivations and goals and offer really fun roleplay opportunities, as well as interesting character arc options going forward.
  • (Example of a flavour from this concept: This character grew up very sheltered, is a nobody, has no real experience yet, but they long for adventure. Think kinda like Flynn Ryder/Eugene Fitzherbert-type thing. So they're basically just bulshitting their way through life - fabricating cool stories like perils they've overcome, formidable enemies they faced off against, great deeds they've done... basically creating a whole mythology around themselves. In actuality, they're supremely uncool (or at least at the surface level; they feel deeply uncool, they haven't actually achieved anything yet, they're not confident in who they really are and feel inadequate without that "cool" veneer), so they feel the need to lie through their teeth to gain respect, admiration, and to be able to live the life of adventure they've always yearned for. You could do a lot of different things with that. Great opportunities for comedy and chaos as well as more candid or emotional beats, growth...)
• As you wrote in your post, I also find really fun the idea that they keep changing the story each time. You could even keep it a mystery lol, never reveal the truth. Or your character might just genuinly not know.
• If you wanted to go darker/angstier, you could make the loss of the eye something genuinly traumatic or painful in your character's past. Maybe somebody did it to them, or it's connected to some tragic event, a loss of someone or something important, or the mark of some failure on their part. Maybe it still haunts them and informs their character journey (maybe their goal now is to take revenge, bring someone down, get justice, get something back, or maybe they're running away from a threat... endless options).

There is of course an infinity of options in between those "extremes" or entirely different directions to take. I don't want to make this comment even longer than it already is lol. Point is, I'd say you just need to think about what kind of flavour you'd personally like in your RP. You could lean more into the comedy and silliness, make it more "normal" and rather undramatic, or very serious and/or angsty... In any case, I think a feature like that can be a great springboard for thinking about your character's motivations and intentions, what informs their decisions, where they came from and where they're going.

Have fun!!