It's possible that he has a different vision for his life. He might have seen a different future for himself, one where he doesn't have to be bound by the rules and isolation of the Night's Watch. He may want to be more involved in the politics and power struggles of the Seven Kingdoms, rather than being on the Wall.
He might go back to Winterfell and claim his rightful place as a Stark. This would change the whole power dynamic in the North and also his relationship with his siblings.
He could meet ancient Valyrian dragonlords. These dragonlords might still have some connection to their dragons, even if the dragons are long gone or in hiding. Another possibility is meeting the descendants of the original Valyrian slaves who have their own unique stories and skills.
Jon Snow could potentially have allies like the Mormonts. They have always been loyal to the Starks. Maybe even some of the Dornish houses might side with him. They have their own reasons to oppose the current power structure in Westeros. And of course, his friends from the Night's Watch who left with him or believe in his cause could be his allies in this conquest. He might also gain the support of some of the maesters who see the value in his leadership for the future of Westeros.
In some fanfictions, he might leave due to a love interest. He could meet someone who makes him question his vows to the Night's Watch. This person could be from a different region or have a special connection to him. For example, she might be a noblewoman with a hidden agenda, but Jon is drawn to her and decides to leave the Night's Watch to be with her and get involved in the political intrigue of the Seven Kingdoms.
Themes could include adventure and discovery. Maybe they embark on a journey together and during that, their relationship develops. There could also be a theme of redemption. If either character has made mistakes or has a troubled past, their relationship could be a means of finding redemption. Additionally, loyalty might be a theme, as they might have to be loyal to each other in the face of various challenges in the fanfiction world.
In a 'Jon Snow is a Vampire' fanfiction, the plot could start with his death and subsequent resurrection as a vampire. This new form could make him an outcast, even more so than he was before. He might have to journey to find a way to reverse his condition. Along the way, he meets old friends and enemies who react differently to his vampiric state. Some might fear him, while others might see him as a powerful ally. His relationships with his Stark siblings could be severely tested as they grapple with the fact that he's no longer fully human.
One possible theme could be power. With Jon Snow having a dragon, it's a huge source of power, and the story might explore how he uses or is corrupted by it. Another theme could be identity. Is he still the same Jon Snow or does the dragon change who he is? Maybe it's about destiny too, as the dragon could be seen as a sign of his fated role in the world of Westeros.
Daenerys Targaryen could be one to help. Since she has experience with dragons, she could give Jon Snow advice on how to care for the egg and hatch it.
It could be a young and untainted Frey. Maybe a niece or a cousin of the more infamous Freys. One who has not been involved in the past atrocities committed by her family and has a more noble heart.
Daenerys Targaryen is often paired with Jon Snow in such fanfictions. Their combined power and the potential for a different kind of relationship makes it a popular choice among writers. Another character is Arya Stark. Their sibling bond can be explored in new ways when Jon is reincarnated, like Arya helping Jon with her assassin skills in a different plot line.