Well when Aang meets the magic turtle he learns all about spirit bending, and how "Before we bent elements, we bent spirits" and then does something to Aang which may only have been some kind of demonstration that unlocked something he could already do, but didn't know about, or may actually have bent his spirit to give him the ability to bend spirits... Which sort of implied something different to me.
So we know that the avatar is reincarnated right? It's the same spirit every time and the spirit has to pass on from the world before it can go back into it and be born into another tribe. What I was thinking is that maybe this was true for pretty much everyone... Here's my hypothetical scenario:
What if way back when in the beginning of time everyone could bend all four elements thanks to the great gods/spirits/whatever... however this proved to be a bit too much power for them, so the spirits in their great wisdom decided to limit everyone's powers to one of the four elements. But then seperating all the people into four groups drives everyone to conflict, so they create the avatar, who is supposed to be a kind of mediator between all the different benders. Time passes on and the four tribes begin to develop into unique cultures, bending passes on through reincarnation of the first people, but since there's a limited pool of bending souls, not everyone is born a bender, this might also lend some explanation to the geneology thing.
I was thinking that maybe the reason why you seem to get these strong bending families that inherit bending despite it seemingly being very random is because bending souls are all snooty and prefer ones that are genelogically more related to them, so more "pureblood" families are more likely to draw a bending soul. Whereas the peasants who seem to have less benders per capita are like that because they've intermixed.