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.
Well, it could be about Arya and Jon's adventures in the unknown lands beyond the Wall. Maybe they encounter strange creatures together, like the White Walkers, and they have to use their combined skills to survive. Arya's skills as a faceless assassin and Jon's leadership and combat abilities would surely come in handy.
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.
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.
The North would surely support him. They have always seen him as one of their own and a great leader. Also, the wildlings might back him, considering he has a history of working with them and understanding their plight. Davos Seaworth would most likely be on his side as well, given his loyalty to Jon in the past.