I'm a fan of odd or unusual player characters. I like creating PCs that could plausibly plug into a typical gameworld in some atypical way, and especially when those PCs explore the edges of what a game system is meant to allow.
Here are two characters that I've thought about playing that show these traits.
The character is a rugged female adventurer with magical powers, suitable for a fantasy game of some kind. Most of the time, one of her eyes is covered with a rough leather eyepatch. During serious fights, she can uncover it to reveal a glowing jewel underneath. The jewel contains the souls of powerful monsters, allowing her to summon or conjure them as controllable pets. Further, if she witnesses certain actual monsters with that eye, or perhaps defeats them, she can acquire them as summons as well.
To me, artificial eyes have always been a very pulp-fantasy type thing, more Howard than Tolkien. Her reason to adventure (travel the world to acquire powerful summons) isn't particularly pulpy or bloody, and doesn't compromise her as a good-aligned hero if that's how I decided to play her. On the other hand, she could also easily be the amoral mercenary magician, conjuring dragons and demons for her own reasons rather than for any larger purpose.
As a player character in an RPG, she covers one of the tricky parts of any game system: the action economy. PCs in combat are typically throttled by the number of actions they can take with respect to other combatants. Characters who can conjure aid, or who have permanent companions, can sometimes gain additional actions. Will the system handle this gracefully? And further, are monsters statted in such a way that summoning and controlling them is viable?
Nick is a superhero character. He's a former police
detective turned private eye. He has amazing psychic powers across
all disciplines - telepathy, ESP, teleportation,
psychokinesis, you name it - but those powers are normally latent.
When Nick is around an object with a strong emotional charge, though, it can activate some of his other powers. For example, a bulletproof jacket that saved someone's life might give him a powerful psychokinetic shield against attacks. A wedding ring that holds the vows of a loving couple might help him find and mentally connect to someone close to him.
As a player character, Nick hits a difficult part of a difficult genre for gaming. How do you create a character who can do a lot of things, pretty well, but one at a time? Many games have some allowances for switching powers, and Nick as a character brings stories into it.
Every object he carries around as a trigger for his abilities has a story behind it, and every object with a story in the game is one that he can use. A lot of supers games basically assume that the civilians in a battle scene are faceless cattle that need to be protected. For a character like Nick, they can be his strongest resource.
Imagine the supervillain has the team battered and beaten in the middle of downtown. The mundanes are cowering… Except there's one kid in the crowd. Nick feels this powerful token on him, tells him to throw it. The kid does. It's a cancer wristband, the kid is a survivor, and the endless hope and fight for life locked up in that thing gives Nick enough biokinetic powers to heal himself and his team back up. Time for Round Two.