I've tried changing armor, travelling, etc., and the arrow is still there. It's really annoying. Any fixes out there for stuff like this?

One trick is to get shot up with arrows again. When those ones disappear, hopefully so will the troublesome ones.

I've seen a ton of reports like this, but hadn't had it happen to me until now. While not on my character, Lydia now has a frost shard stuck in her arm. I'm hoping it's gone the next time I load to continue my adventures.

I imagine you're playing on PC with your signature so there is actually an easy fix. Open the console with ~ and type sixchange. It will make the arrows disappear, just type it once more now to look normal again

Don't worry too much about the arrow, just find some bandits and let them pepper you with arrows until the problem is gone. Worked for me first try. PS3 user if that makes any difference to you.

As to you Ohden, I suppose attempting to find bandits using the frost shard spell may be quite difficult, you could save and try shooting on or two in her yourself and seeing if it will remove the bugged one.

Had that for a while. Unequip and drop/store all arrows on your character.

Think about when you shoot enemies and see the arrows sticking out of them, they disappear when you remove the arrow from their inventory, so do that.