The story would shift in terms of the overall power balance. With a dragon, Jon Snow could play a more active role in the politics of Westeros. He could influence the decisions of other lords more easily. He might also be more involved in exploring the mysteries of the world, such as the secrets beyond the Wall. The dragon could open up new storylines, like Jon Snow using it to search for ancient magic or lost Targaryen treasures. This would add a whole new dimension to the fanfiction, making it full of new possibilities and adventures.
In this fanfiction, the relationship between Jon Snow and other characters would be different. For instance, Sansa might view him as an even greater asset to the North. Theon would likely be in awe of him. His relationship with Arya could also be affected as they might plan new adventures together with the help of the dragon. And his enemies, like Cersei, would be more desperate to get rid of him as his dragon - powered presence would be a huge threat to her rule.
In this fanfiction scenario, Jon Snow hatching a dragon would completely transform his relationship with Daenerys. At first, she might be in awe of the fact that Jon, a man she is already attracted to, has this new and powerful connection with a dragon. But as time goes on, power dynamics could shift. The dragon could act as a sort of mediator between them, or it could drive a wedge. If the dragon has a will of its own and favors Jon over Daenerys, it could cause a rift. However, if they can learn to work together with their dragons, they could become an unstoppable force in the fight against their common enemies.
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.
Well, with the addition of dragons in the fanfictions, Jon Snow can be seen as more of a magical or otherworldly figure. He starts to incorporate the power of the dragons into his identity. He might become more solitary too, as the power of the dragons sets him apart from others. He has to figure out how to control and use this newfound power, which changes his decision - making process. He may be more willing to make bold moves that he wouldn't have considered before, like flying into enemy territory on the back of a dragon to strike fear into his foes.
Perhaps in a fanfiction, Jon Snow gets a castle that has a hidden magic. This magic could be the key to uniting the Seven Kingdoms in a more peaceful way than through war. He discovers ancient secrets within the castle walls that help him in his quest for a better realm.
Their relationship might become more equal. Previously, Jon was more of an older brother figure. But now, with Arya's newfound skills and experiences, they would see each other as equals. They would respect each other's abilities more deeply.
As a Faceless Man, Jon Snow could have a different role in the fight against the White Walkers. Instead of leading an army openly, he could use his shape - shifting abilities to spy on the enemy. He could blend in with the White Walkers' ranks and find out their weaknesses, which would be a huge advantage for the living.
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.
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.
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.
One implication could be a shift in power dynamics. If Jon Snow has Blackfyre, he becomes a more formidable force, which might make other characters either fear or seek to ally with him.