Purely from habit, I always remove those annoying stringy things whenever I peel a banana. I'd never really thought about it before, just assuming they were inedible, or at least something I didn't want to eat. As it turns out, they have a name - "phloem bundles" - an essential part of the banana's vascular system that delivers nutrients throughout the fruit. If you don't mind their tough texture, you can absolutely eat them. As for me, I prefer not to have strings in my banana bread, but you sure can leave them in if you like.