A common Avatar: The Last Airbender theory suggests Katara would've become the Avatar if Aang never went into the ice. At the outset of the Hundred Year War, Aang took Appa and ran away from his home in order to avoid being sent to another Air Temple, where he would continue his training as an airbender. Getting caught in a storm, Aang plunged into the water and unwittingly entered the Avatar State. Doing so protected him and Appa, and the iceberg Aang created kept them alive for a hundred years.
In that time, the Avatar Spirit never transferred to anyone else, because Aang hadn't died. He was still the Avatar, and no matter what Fire Lord Sozin did to the Air Nomads and, later, the Water Tribes, the Avatar still lived. It was only after Aang passed away decades later that the Avatar Spirit was reincarnated in Korra, a waterbender. But looking at the timeline, what if Aang never went into the ice and died at as an old man like his friend, Bumi? Would Katara have become the Avatar?
Because Aang was the Avatar, he led a hard life that ultimately took its toll on him, at least compared to someone like Bumi, who lived to be well over 100 years old. But if Aang could've held out until he was near 100, then the Avatar Spirit might've transferred to Katara since she's a waterbender from the Southern Water Tribe (like Korra), which wasn't too common. Plus, she was one of the most skilled waterbenders in the series, so there's a level of inherent skill there.
The timeline works out well with this theory, but it, of course, relies entirely on the timeline falling into place. If Aang lived a shorter or longer life, then Katara wouldn't have been the Avatar, nor would have Korra. It could've been another waterbender, such as someone from the Northern Water Tribe. It's interesting to think about, though; if Aang had thwarted the Hundred Year War and defeated the Fire Nation, Katara's time as the Avatar would've looked entirely different than what fans saw in the series. There may not have been a global war and she could've spent her life maintaining the peace that Aang forged.
Another person who could've been the Avatar, though, is Pakku, if Aang had failed in his mission to defeat the Fire Nation and ultimately died at a young age. Pakku, who also has ties to Katara, was trained by waterbending masters and became a member of the White Lotus later in life, so it's in his blood to work with and support the Avatar. Although the Avatar could really be anyone - doesn't have to be a named character - he's someone who has the necessary skill to pull it off. Again, the entire theory is reliant on the timeline and when Aang would've died, but the idea of the Avatar Spirit reincarnating for thousands of years opens up the story to numerous possibilities and "what if" scenarios, especially given how much time Aang was in the ice.