Obviously, Jon Snow is a main character. He's usually depicted as having a special bond with the ice dragon. His character traits like his honor and leadership often play a big role in the story.
In some fanfics, other characters from the 'Game of Thrones' universe can also be important. For example, Daenerys Targaryen might make an appearance, especially if the story is about the relationship between the ice dragon and her dragons. Her knowledge of dragons could add an interesting dynamic to the story as she tries to understand the ice dragon that Jon has bonded with.
The ice dragon itself is also a main character. It might be portrayed as a powerful, mysterious creature. Sometimes it has a will of its own and other times it seems to be in tune with Jon's desires.
One great 'Jon Snow Ice Dragon' fanfic could be 'The Dragon's Bond'. In this fic, Jon Snow discovers a hidden connection with an ice dragon. It's full of adventure as he tries to understand this new relationship and use it to protect the North.
Well, the dragon would obviously be a main character in its own right. It could have its own personality and motives.
White Walkers can be enemies in some fanfictions. Even though the ice dragon is a powerful ice - based creature, the White Walkers might see it as a threat or something they can control. Another enemy could be rival dragon riders. They may envy Jon Snow's ice dragon and try to either steal it or defeat it in battle.
Obviously, Jon Snow as a reborn dragon is the main character. But there could also be characters like Daenerys who might play an important role. She has her own connection with dragons, so her reaction to Jon's new form could be a significant part of the story.
One popular 'Jon Snow finds an ice dragon' fanfiction might be 'The Ice Dragon's Bond' where Jon Snow forms a deep connection with the ice dragon. The dragon becomes his loyal companion and helps him in his battles against the White Walkers. It shows how their bond changes the course of events in Westeros.
Sansa Stark might be involved as well. Given her relationship with Jon Snow as his sister (though later revealed otherwise), she could play a role in protecting or being involved with the baby. Maybe she helps raise the baby in a world where Jon Snow is constantly facing threats.
A common theme is power. Jon Snow's transformation into a dragon gives him a new level of power, which often leads to a change in his status within the story's world. Another theme is identity. He has to grapple with being part of a new species, so to speak.
Jon Snow could interact with the dragon egg through his dreams. In the fanfic, the egg might communicate with him on a spiritual level while he sleeps. These dreams could give him clues about the egg's power and what he is supposed to do with it. Also, he may touch the egg and feel a jolt of energy, which makes him realize that there is something truly special about it.
Often, these fanfics might feature multiple love interests for Jon Snow and complex romantic storylines.
Fanfiction.net is another place to search. There are a lot of fans who post their 'Jon Snow finds an ice dragon' fanfictions on this platform. Just use the search function and type in relevant keywords to find what you're looking for.