The only way I would know to do this is to leave the effects of the customization up to the DM. The rules would be unchanged, but the effects would have to be ruled on by the DM in every situation.
For example, we (a group of about 8-10 of us gamers) used to play a little homebrewed game for fun in the car (on the 2 hour trip to the big city to hit the game shops). It went something like this: each person thought of a character. Characters had a name, a race, a career, two talents and three tricks. And one "prop". There were no limits to what any of these things could be, except that they were limited to a pair of words or short phrase.
My favorite character was: Human Female, circus acrobat. Talents: horseback riding, gymnastics. Tricks: disguise, mimicry and pilfering. Prop: umbrella.
Talents added +2 to any applicable d20 roll. Tricks added +1. Props gave anywhere from +5 to +1, or simply allowed success, at DM's discretion.
The game was totally narrative, and we traded DM roles around the group every 15-30 minutes. The only rule was that if you wanted to roll off for success, high roll always won. Ties went to whoever made the other person laugh harder.
The game depended absolutely on trust in the DM, but it was also totally flexible, because if you didn't like a DM's decisions, you just waited til you were DM and changed them. Very fun, but not very suited to "campaigns".