I personally think a story where the Avatar has a dark opposite like an Anti Avatar would be good or a story where there are two Avatars like twin brothers born and both possessing the powers of an Avatar but in constant conflict with each other.
The reason I think this could work with Twins is because the Avatar is always reincarnated but what would happen if the Avatar had a twin? I mean would that mean that the Avatar split into two and what impact would that have? Would they both be good or be destined to be enemies and if the reincarnation bit still applies would that mean the two avatars would be destined to always fight each other throughout their new lives?