One simple way in fanfiction could be through some kind of magic. Maybe there's a powerful red priestess who discovers Ned's body before it's fully decomposed. She uses her magic to resurrect him, much like Thoros of Myr did with Beric Dondarrion. Once back, Ned could play a major role in the political games again, this time with more knowledge and a thirst for revenge against those who wronged him and his family.
Another fanfiction could be about Jon Snow (who is later revealed to be Ned Stark's son). In this story, he could fully embrace his Targaryen heritage much earlier. He might form an alliance with Daenerys not as an equal but as a leader, using his knowledge of the North and the loyalty of the Night's Watch to help her claim the Iron Throne in a more peaceful way.
One interesting storyline could be Ned Stark ruling justly in King's Landing. He might reform the corrupt systems in place. For example, he could ensure fair trials and end the excessive use of power by the nobles. This would likely earn him both respect and enemies among the powerful families. He could also focus on strengthening the city's defenses and improving the living conditions of the smallfolk, which would be a departure from the self - serving rule of many previous kings.
In the fanfiction world, Ned Stark might kill Littlefinger if he discovers Littlefinger's true manipulative nature early on. Maybe he finds proof of Littlefinger's role in some of the political schemes that are endangering his family. Ned is an honorable man, and once he has evidence of such treachery, he would not hesitate to dispense justice.
A plot twist could be that Ned Stark's son Arya (disguised as a boy for most of her journey) is actually a secret weapon trained by the Faceless Men not just for personal vengeance but to be a political pawn. She could be used by some powerful force to manipulate the balance of power among the kingdoms.
He'd become more politically astute. As a king, Ned would have to deal with the complex web of politics in King's Landing. He couldn't just rely on his sense of honor. He'd need to learn to play the game, which would be a big change from his straightforward nature in the North.
Arya Stark is one of the most beloved characters from 'Game of Thrones'. She is initially presented as a bit of a tomboy, not conforming to the ladylike standards of her society. As the story progresses, she becomes a key player in the power struggles of Westeros. She is brave, resourceful and has a very distinct personality that makes her stand out among the many characters in the series.
One great fanfic could be about Arya Stark rebuilding Winterfell as the last Stark. It might explore how she uses her skills she learned during her travels to bring back the glory of her family's home. She could train new guards, establish trade routes, and make Winterfell a stronghold again.
In some fanfics, Arya Stark might be presented as the last Stark, but it really depends on the specific plot and direction the fanfic author takes. There could be plot twists where other Starks are revealed to still be alive.