In this fanfiction, it could be that the dragon is actually a key to unlocking some ancient magic or prophecy. Jon Snow, finding the dragon beyond the Wall, realizes its significance. He teams up with the dragon, and together they embark on a journey to find out more about this magic. Along the way, they face many challenges, both from the natural elements beyond the Wall and from the political intrigue brewing in the Seven Kingdoms as word of the dragon spreads.
The White Walkers could be the main antagonists. Since Jon Snow has found a dragon, they might see it as a threat to their plans of taking over the Seven Kingdoms. They would try to kill the dragon and Jon Snow to ensure their dominance remains unchallenged.
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.
The wildlings might be his allies. After all, Jon Snow has had interactions with them before. They are fierce fighters and could fight alongside him against a common enemy.
One aspect that fanfiction might focus on is how Val's wildling background interacts with Jon Snow's upbringing in the Night's Watch. Their cultural differences could lead to both conflict and a deeper understanding between them. Maybe in some fanfics, Val helps Jon Snow better understand the ways of the wildlings, which in turn gives him an edge in dealing with the threats beyond the Wall.
In this fanfiction, one of the key battles Jon Snow might participate in could be the Battle of the Whispering Wood. His skills as a fighter would be put to good use there, sneaking around and taking out enemies. Another could be a fictional battle near the Wall, where he defends against some new threat that has emerged during Robb's war.
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.
A great deal of Jon Snow and Iron Throne fanfictions focus on his relationships. For example, his relationship with Daenerys could be very different in fanfictions. In some, they rule together on the Iron Throne, combining their claims and powers. In others, they are at odds, with their relationship being a major obstacle in Jon's path to the throne. And of course, there are fanfictions where his Stark family ties also influence his actions regarding the Iron Throne, as loyalty to his family is a strong part of his character.
One idea could be a crossover where Jon Snow is transported to the Naruto universe. He could join Naruto's team and use his combat skills from Westeros, like sword fighting and archery, in the ninja battles. His leadership skills might also be recognized in the Hidden Leaf Village.
One idea could be a story where Robert Baratheon discovers Jon Snow's true parentage earlier. This could lead to a complex relationship where he sees Jon as a potential heir or a powerful ally. He might take Jon under his wing, train him in the ways of leadership and warfare, much like he did with Ned Stark's other children.