Perhaps in the fanfic, Jon is a dragon trainer in disguise. He finds a dragon that has been lost from its group. He uses his skills to communicate with the dragon and tries to reunite it with its kind. Along the way, they face many challenges and obstacles together.
In this fanfiction, it could be that the dragon is actually a key to unlocking some ancient magic or prophecy. Jon Snow, finding the dragon beyond the Wall, realizes its significance. He teams up with the dragon, and together they embark on a journey to find out more about this magic. Along the way, they face many challenges, both from the natural elements beyond the Wall and from the political intrigue brewing in the Seven Kingdoms as word of the dragon spreads.
Maybe it's about Jon finding a dragon egg and through some magical or difficult process, he manages to hatch it. He could face challenges like protecting the egg from those who want to use it for evil, and once the dragon is hatched, they form a strong bond and go on adventures together.
Jon Snow could discover that the dragon has some sort of special connection to his true heritage. Maybe it was left there by his Targaryen ancestors. The dragon could help him uncover secrets about his family and his place in the world. He may also face challenges in controlling the dragon at first, but with time, he masters it and uses it to change the fate of Westeros.
In the fanfic, Jon might use the dragon egg to gain power. He could try to hatch it and form an alliance with the dragon. This would change the power dynamics in Westeros completely. Maybe he'd face opposition from other powerful houses who fear his newfound strength.
Well, in the fanfic, it could be that Jon Snow stumbles upon the dragon egg while on a mission beyond the Wall. He might not even realize what it is at first. But as he holds it, some latent magic within him starts to interact with the egg. As it hatches, the dragon imprints on Jon like in some old legends. The dragon's presence then forces Jon to reevaluate his role in the world. He could use the dragon to fight the White Walkers more effectively, or he might face opposition from the current rulers who fear his new - found power. It could also open up a whole new arc where Jon has to learn how to train and communicate with his dragon.
Well, in this fanfiction, it's possible that the dragon egg has some special connection to Jon Snow's true parentage. It could be a sign of his Targaryen heritage, if we follow the theories. He could then use the dragon that hatches from the egg to fight against the White Walkers, as dragons are very powerful creatures in the 'Game of Thrones' universe.
Another option could be Fanfiction.net. Sign up or log in if you have an account, and then search for the specific 'jon finds dragon' fanfic. However, the tagging system might not be as detailed as AO3, so you might need to sift through a few more results. But it's still a great place to start looking for fanfics.
Well, perhaps Jon finds a dragon egg in a hidden cave. He takes it home and starts to take care of it. As he does, he begins to notice strange markings on the egg that seem to glow at night. Eventually, the egg hatches and out comes a small, cute dragon that he names something like 'Spark'.
One possible plot twist could be that Jon Snow, as a dragon, has to hide his true form from his former friends in the Night's Watch. They might see him as a threat now and he has to struggle to prove that he's still on their side, even in his new form.
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.