Well, it depends on the specific fanfiction. Some fanfictions do give Jon Snow dragon magic. They might do this to add an extra layer of mystery and power to his character. Since he already has a complex backstory in the original series, the addition of dragon magic can make him even more of a central and powerful figure in the fan - created stories.
Well, in a lot of fanfiction, Jon Snow's dragon magic is a very powerful force. It might be that he can use it to heal the dragons if they are injured. Or he could use it to enhance his own combat abilities, like summoning dragon - like fire or strength. This not only makes him a more formidable character but also allows for new and interesting interactions with other characters in the fan - created universes.
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.
There might be a fanfiction where Jon Snow's magic is related to his connection with the White Walkers. Instead of just being a fighter against them, his magic allows him to understand their language or even control some of their actions. This would create a very complex and unique plot as he tries to balance this new power with his loyalty to the Night's Watch and the North.
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.
Daenerys Targaryen is often paired with Jon Snow in these fanfictions. Their connection in the original series, along with their both being strong leaders, makes for an interesting dynamic. They could rule Westeros together in a new era of peace and prosperity after Jon's rebirth.
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.