As for what I'd base it on...
I'd suggest characters be built out of three broad parts: archetype (class), background, and then a handful of merits.
I would drop race/ancestry/heritage entirely - as things stand, they're little more than a package of minor powers, and they're not all appropriate to all settings anyway. So instead I'd provide various merits for the most common abilities (low-light vision, for instance), and allow groups to describe them however they consider appropriate.
I think I'd also be inclined to drop ability scores entirely. The existing six are one of the genuine sacred cows of D&D, but the boundaries between them have always been debatable, and anyway they're largely chosen to optimize for a class anyway. So, again, I'd offer merits like Very Strong, Eidetic Memory, and so on as replacements.
In terms of resolution mechanic, I'm actually entirely agnostic - ultimately, it all boils down to a way to determine how good a character is at a given task, and therefore the chance of success against level-appropriate challenges: Perfect (approx. 100% chance of success), Good (approx. 70%), Average (approx. 50%), or Poor (approx. 20%). But there are lots of ways to skin that cat.