I would say number 2:
Being an avatar is purely spiritual, and cannot be 'passed on' as with normal bending genes. This may imply however that non-avatars can become avatars if they contemplate their navels thoroughly enough...
I believe this because as of yet, we have no information that Korra is related to Aang, nor that Roku or any of the other past avatars are related to each other.
Also given the fact that there is a new avatar only at the correct time, ie the old one is always gone before the new one arrives, I would say that 'passing on' the avatar status is spiritual and not genetic.
However, I do not beleive that "non-avatars can become avatars if they contemplate their navels thoroughly enough".
The avatar is a spiritual thing, the avatar is the bridge between the spirit world and the human world. The Avatar Wiki backs me up on this:
The Avatar is the spirit of the planet incarnated in human form
The world is an entity, entities only have one spirit, Therefore there is only one bridge between the spirit world and the earthen world at a time.
One world, one avatar, that is all there can be.
An interesting possibility is that there are many potential avatars, humans who have the avatar ability within them, simply lying in a dormant state. Similar to Potential Slayers in Buffy.
Now how the spirits choose which human to unlock the avatar power for, if they even choose at all for that matter, is mostly a mystery as far as we know. There are some simple guidelines the choosing of the next avatar follow however:
- The new Avatar must follow the Elemental Cycle: Fire -> Air -> Water -> Earth -> Fire etc.
- The new Avatar should be strong spirited and have good enthusiasm.
There is only ever one avatar at a time, however there could very well be mulitple potential avatars, ready to be called upon by the spirits should the current avatar expire early.