I wouldn't like a complete, full background, but something a little more specific, just a single element of your background that could be spun off into a story thread / quest line / sub plot. I don't want a full Origin-style backstory (nor do I want to play those characters), I want to create my own character; but having something that anchors them a little, that gives them a past reflected in the game, would be nice.

Honeslty, I think part of the problem, as I've written elsewhere, is that your companion characters are actually protagonists who all have their own stories going on. This makes your character seem like more of a cypher by comparison.