We're already given two choices for the player: either a heavily scripted Origin character or a blank slate for roleplaying.
A middle option with some kind of "light" personal quest would be hard to pull off without steering too far in one of those two directions. Too much scripting and it would feel like your story is on rails, not much different from choosing an Origin character. Too little would just be a "meh" side quest with no real impact. That's without even getting into how it would have to fit each race and class.
It's an interesting idea, I just don't think it's practical, considering the opportunity cost in developer time that could go somewhere else.
