If they were only going to pick ONE of these to key your custom character's story off of, would you prefer:
1. Your class.
2. Your race.
3. Your background.
Which one of those would you want the most to be the main determinant of what kind of storyline/origin they make for your character?
For me, it would be background. As I think you could do a lot more interesting things with like Charlatan, Urchin, Criminal, Acolyte, etc. than with the others.
I would say none of the above. Make the PC Origin more general, allowing the player to pick through dialogue what applies to their PC. No need for a dizzying array of options that would take tons of time to code all the permutations. 1-3 solid PC choices for a custom origin would be great (with quests, story, etc...)