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.
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.
In some fanfictions, if Jon Snow goes to Old Valyria, he may meet characters from the past. He could interact with ancient Valyrian families or even those who were involved in the events that led to the Doom. This could give him insights into the history of the world and how it has shaped the present, and potentially give him ideas on how to deal with the threats in Westeros.
In this fanfiction, Jon Snow's character could develop in several ways. Firstly, his relationships with other characters might change. He may form stronger bonds with his fellow soldiers, learning to trust them more. Secondly, his understanding of politics and power could grow. As he is involved in Robb's war, he witnesses the political maneuverings that go on behind the scenes. This could make him more cynical or more determined to fight for justice, depending on how he interprets these events. Moreover, his combat skills are likely to improve, and this newfound strength could give him a different perspective on his own capabilities and his place in the world.
One possible plot twist could be that Jon Snow goes back in time and accidentally changes something that leads to a completely different future for the Stark family. For example, he might prevent Ned Stark from going to King's Landing, which would change the power dynamics in Westeros entirely.
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.
His relationship with his father, Ned Stark, would be different. He could finally have a real father - son relationship instead of the hidden one. He might be able to learn from Ned and become an even better leader in the North.
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.
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.
Fanfiction about Jon Snow and the white dragon can go in many directions. Some stories might explore the origin of the white dragon and how it came to cross paths with Jon. It could be that the dragon was hidden away for centuries and Jon's arrival in a certain place awakens it. The relationship between them could also be one of mutual respect and understanding, with the dragon seeing something special in Jon that makes it want to follow him, like a kind of destiny or prophecy at work.