I'm not talking about "My character eats and sleeps even though they don't have to" or "My character doesn't use magic because he doesn't trust it" -- those are common RP/character choices to make. Even something like "My character has to drink wine all day because he's an alcoholic," doesn't really count because it's a rational character choice.
I'm talking about bizarre, strange things. Maybe your character has to drink a Honningbrew Mead whenever she's in Whiterun to raise a toast to the Headless Horseman? Or maybe your character has to punch a wolf in the face before going into a bandit camp (for good luck)? Maybe your character refuses to enter a Dwemer Ruin without first taking off all his clothes? The possibilities for strangeness are endless.
My character only wears an Iron Helmet in battle, because it was her first piece of armor and she -- and I -- have an irrationally strong attachment to it.
Also, every time she enters a Jarl's palace, she has to knock everything off the banquet table. Without exception.