Yes, there are quite a few. Some explore how Jon Snow might bond with the dragon, given his unique position in the story. For example, in some fanfictions, his connection to the dragon is seen as a sign of his true Targaryen heritage coming to the surface. The dragon becomes not just a powerful weapon but also a symbol of his destiny.
One possible plot could be that Jon Snow discovers the dragon egg in a hidden place. He manages to hatch it through some unforeseen circumstances. Then, he has to keep it a secret while training it and learning to communicate with it.
Perhaps the dragon in Jon's fanfiction is the last of its kind, and Jon has to protect it from those who want to harm it. There could be a whole story of them hiding, fighting off hunters, and trying to find a new place for the dragon to live safely.
Finding 'Good Jon has a dragon' fanfiction can be a bit of a hunt. Start with the big fanfiction platforms. Archive of Our Own has a large collection of various fanfictions. You can use their search function to look for it by the title or relevant tags. FanFiction.net is also a well - known place. If it's related to a specific franchise, look for fan communities around that franchise. For example, if it's related to a book series, check out the book's official forum or any large - scale fan - run forums. There might be someone who has come across this particular fanfiction and can point you in the right direction.
One possible storyline is that Jon Snow uses the ice dragon to fight against the White Walkers. The ice dragon's cold breath could be a powerful weapon against the undead army. Another storyline could be that the ice dragon helps Jon Snow claim the Iron Throne. Its presence would make him a very intimidating and powerful claimant.
There are so many dragon - related fanfictions out there. Maybe you could look at the tags on the fanfiction site where Jon's story is posted. If it has tags like 'fantasy dragons' or 'dragon adventures', search for other stories with those same tags. That way you might find some that are in the same vein as Jon's dragon fanfiction.
There could be a wise old wizard who gives Jon advice on how to hatch and take care of the dragon. Maybe there's also a rival of Jon's who is jealous of his discovery and tries to steal the dragon egg.
If Jon Snow gets a dragon in fanfiction, he might face new challenges. People would be both in awe and envious of his newfound power. There could be plots against him not just from political rivals in Westeros but also from those who fear the power of dragons. He would have to learn how to control the dragon quickly. Given his noble nature, he might try to use the dragon's power for good, like protecting the smallfolk and rebuilding the war - torn regions.
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.