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.
It's highly unlikely that the characters in 'House of the Dragon' would actually watch fanfiction about themselves. Fanfiction exists in our world, created by fans for fans, and the characters in the show are fictional entities within their own fictional universe. They don't have access to the fan - created stories about them.
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.
One common theme is power. Since dragons are often associated with great power in fantasy, Jon's relationship with the dragon can lead to him gaining new powers or learning how to use power responsibly.
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.
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.
One idea could be that it's a result of some ancient magic in Westeros. Maybe there are hidden spells or prophecies that allow for such a transformation. In the fanfiction, this could lead to Jon having new powers like breathing fire and flying, which would completely change the power dynamics in the story.