Question: Element Benders can pass on their abilities genetically. I'm wondering is an avatar that has access to more elements than usual is able to pass on more than their native bending talent. We haven't, to my knowledge much to go on but this family tree.

![enter image description here](https://i.stack.imgur.com/I76q1.jpg)

I doubt we'll see any hard evidence soon (i.e. proof by contradiction), but if there is something I've missed theory wise, it would help answer.

I see two lines of thought on this:

1. Being an avatar is if not hereditary, some how genetic. Some portion of gene could be passed on unlocking non-native bending.
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...

Which one is correct, or is there a third option?

Answer: 
TLDR; No -- because the Avatar doesn't have all the bending genes.
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The Avatar does not have the genes to bend all the elements; rather, as shown in [The Legend of Korra](http://avatar.wikia.com/wiki/Beginnings,_Part_1), the Avatar's multi-bending is due to his or her spirit being merged with [Raava, the Spirit of Light](http://avatar.wikia.com/wiki/Raava).

![Raava](https://vignette2.wikia.nocookie.net/avatar/images/f/f5/Weakened_Raava.png)

Raava, being a spirit, is able to hold multiple bending arts at once. Human spirits, on the other hand, can only hold one. The merger of the first Avatar, [Wan](http://avatar.wikia.com/wiki/Wan), with Raava allowed him and his reincarnations to bend multiple elements at once.

On the interesting end of things, the state of the Avatar franchise in the first two series had people who self-segregated into nations by the predominant bending ability. This assured that any Avatar was born to a member of a tribe that was known for a type of bending.  

![FourNations](https://vignette1.wikia.nocookie.net/avatar/images/b/b7/Avatar_World_map.png)

From that point, we see that the Avatar will not only be merged with the spirit of Raava, but will also have the genes of a bender in their own right, (incidentally, it's impossible to tell if they're currently expressed, as the Avatar's bending comes from Raava) which can be passed on to their children. They cannot, however, pass on the genes they do not have.

Eventually, however, all the bloodlines will mix as they have begun to in Korra (the nations, now at peace, have started fusing and their people intermixing), and all people will have the potential to be born benders of any single variety.
No, the avatar can only pass on their native elements to their children. Hope that helps. :)