Luck played a major role in Aang surviving the Air Nomad genocide. If he had not chosen that exact moment to leave the Southern Air Temple, he more than likely would have been killed during the Fire Nation attack. Even more luck was involved to cause Aang to get caught in a storm and freeze himself for a hundred years. If these events didn't play out as they did, the world of Avatar would be drastically different.
But what if Aang hadn't frozen himself and lived out his natural life span during the Hundred Year War? Once he finally died, who would have become the next Avatar? It is known that after air, the next Avatar is born into the Water Tribe as a waterbender. With both the North and South Poles as possibilities, numerous waterbenders could have been chosen to be the next Avatar.
Yue
It's a popular theory among fans that Yue was meant to be the next Avatar after Aang if he had lived a natural lifespan. When Yue was born, she had an illness that would have killed her if her parents had not placed her in the Spirit Oasis. The Moon Spirit gave Yue a part of its life force, preventing her death. It's possible that Yue's spirit wasn't complete because she was meant to hold the Avatar spirit.
Whether or not Yue was destined to become the Avatar, she more than likely would have excelled at the position. She was entuned with the spirits and showed reverence to them. She also took her duties as a princess seriously. Yue was willing to enter into an arranged marriage for the good of her people. She even sacrificed her life to save the Moon Spirit and potentially the world. Yue brought balance to the world just as an Avatar would.
Although she never showed any waterbending abilities during her lifetime, when she became the Moon Spirit, she showed impressive skill, creating a massive tidal wave that safely carried Aang to shore when he was in danger. If Yue had received the Avatar spirit, she would have been an incredibly powerful bender.
Katara
Legend of Korra revealed that the Avatar after Aang would be born a girl in the Southern Water Tribe. Katara fits those requirements perfectly. It's possible that if Aang had lived and never been frozen, she would have been next in line. Katara did end up becoming Aang's waterbending teacher and later married him. She has had a close relationship with the Avatar, and her grandmother said that their destinies were intertwined. Maybe Katara was drawn to Aang because she was meant to be the Avatar herself.
Once Katara received proper training, she showed phenomenal waterbending skills; indeed, many consider her to be the strongest waterbender in the series. Katara could be such a powerful waterbender because she was destined to be the Avatar. She definitely has the skill to fulfill the job. Combining Katara's natural skill at waterbending with the power of the Avatar would make her unstoppable.
Pakku
Aang almost died in the Air Nomad genocide. If he had, the title of Avatar would have passed onto the older generation of waterbenders. The waterbending master Pakku could have been next in line to be the Avatar if Aang had died early in the Hundred Year War. Pakku was always a gifted waterbender and became a master at a young age, and his abilities would have greatly helped him as the Avatar.
Just like Katara, Pakku also trained Aang, so his fate was likewise intertwined with the Avatar. Pakku also fought to restore balance to the world as the Avatar would. During the Seige of the North, Pakku fought to protect the Northern Water Tribe. He later went on a relief mission to the Southern Water Tribe to help rebuild the nation. He also worked with the White Lotus to liberate Ba Sing Se.
Pakku has both the mindset and the skill to be an effective Avatar. Despite his outdated ideas about gender, Pakku would have worked to protect all people of the four nations. If the mantle of Avatar had been passed down to him, the world would have been in safe hands.
Hama
Hama is another female waterbender born in the Southern Water Tribe that the Avatar spirit could have passed to. The Fire Nation started raiding the Southern Water Tribe in the hope of capturing or killing the new Avatar. If Hama had become the Avatar after Aang, they would have succeeded. Like Katara and Pakku, Hama was a powerful waterbender well suited for the title of Avatar.
Hama would have been the first known Avatar to learn bloodbending. This new Avatar would have behaved in a much darker manner than previous Avatars. Hama would have pursued a violent path that would have been very different from Aang's passivism. Both the Fire Nation and possibly the rest of the world are lucky that Hama never became the Avatar.