hmmmmm this is a very intriguing question
You can go by house or you can go by character (Targs would be fire benders either way, that or non-benders: hello Viserys)
I love love love the idea of Rhaegar being the avatar and after his death no one knowing who the next one is
The Night’s Watch is comprised of all kinds of benders (water benders helped build and maintain the wall). They also had a high number of non-benders specializing in blocking bending, though that is now a dying art. Wildlings have mostly air and water benders… Which makes the Wall slightly more interesting and complicated (perhaps it acts as a barrier against bending, you can’t bend while being in contact with it, because of maaagic).
I can see the Starks being mostly water benders, however they get it from their mother(Tullys are water benders). This goes against what we know of the water tribe living where it’s cold etc but I just really fancy Ned, Arya and Jon being earth benders and Arya becoming a metal bender(even though they have a sword named ice, whatever). Sansa is definitely a water bender who learns to heal, Bran can move plants by using the water inside of them and access their memories, Rickon is a natural bloodbender and Robb is just a run of the mill water bender (sorry Robb).
The Baratheons are generally earth benders, Robert was a metal bender and Renly was a non-bender a fact which made Stannis even angrier that Robert gave him Storm’s End. Robert’s bastards are all exceptional earth benders and Gendry is a metal bender.
Martells are air benders, with a few fire benders popping up here and there. Doran is an air bender and while it’s known that air bending is mostly a defensive style, air can make a fire much stronger or snuff it out completely.
The Lannisters are non-benders. They don’t need bending they have money. Tywin married off Cersei (and was planning to marry her to the Avatar) so there could finally be a bender in the family. It would also not make it immediate apparent but still questionable that none of Cersei’s children can bend and add another layer to Joffrey’s daddy issues.
Tyrells as water benders, cause they just strike me as such. Loras can bloodbend without thinking and Margaery has a more calming, nurturing approach to her bending. Garland and Olenna don’t bend and Willas used to be a very powerful bender until Oberyn accidentally hurt him and also blocked his bending.
The Greyjoys could be air benders, despite the more obvious choice. After all, drowning has more to do with lack of air than it has with the presence of water. Also the supposed horn that exists makes me want to call them air benders even more. (I still haven’t convinced myself they shouldn’t be water benders btw but I like them as air benders.)
Melisandre is in contact with the spirit world and shows signs of being a fire bender. Davos is a water bender, he used mist to smuggle and it was how he managed to escape death on the Blackwater. Arryn is obviously an air bender… and that is all I can think of right now.
EDIT: forgot to mention that all wargs use energy/spirit bending to enter another body also more headcanons here
Game of Thrones and Avatar crossover. Beautiful