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.
In this fanfic, Jon Snow could be a special Pokemon ranger. He is sent on a mission to a region that has been affected by a strange phenomenon that is making Pokemon act aggressively. Jon, with his experience in handling difficult situations in Westeros, uses his wisdom and combat skills. He forms a bond with a Growlithe, which he names Ghost after his direwolf. Together, they work to solve the mystery and restore peace to the Pokemon region.
One great baby Jon Snow fanfiction might be 'The Young Wolf's Beginnings'. It could explore his early days in Winterfell, his relationship with his family members like Ned Stark. Another could be 'Jon Snow: Infant of Prophecy' which delves into how his birth was a significant event even from a very young age. And 'Baby Jon's First Winter' might focus on his first experiences of the harsh winters in the North.
You can try fanfiction websites like Archive of Our Own (AO3). It has a large collection of fan - created works, and there are likely many baby Jon Snow fanfictions there. Just use the search function and type in relevant keywords like 'baby Jon Snow'.
One interesting 'jon snow baby fanfiction' could be about Jon Snow having a baby with a character from another house, and the story revolves around protecting the baby from the political intrigue in Westeros. Maybe the baby has some special powers that make it a target.
The White Walkers could be antagonists. They pose a threat to everyone in Westeros, including Jon, Ygritte and their baby. Their icy presence could endanger the baby's life and force Jon and Ygritte to fight for its survival.
Tyrion Lannister could be involved. He was close to Daenerys and had a relationship with Jon as well. He might be the one to help protect the baby or offer his wisdom in raising the child. Another character could be Ser Davos. He was loyal to Jon and would surely be interested in safeguarding the future of Jon's heir.
It depends on the specific fanfic. There are countless fanfictions out there, and while some may choose to give Jon Snow dragons to enhance his character or create a more fantastical story, others may stick closer to the original lore and not include such a plot element. So, you really have to dig into different fanfics to find out if he has dragons in them.
In a fanfic, Jon Snow could be called a bastard during a tournament. Rivals from other houses might use it to try and get under his skin before a joust or a melee. Also, in a tense council meeting where his status and right to be part of important decisions were being questioned, the term 'bastard' could be thrown at him. Another instance might be when he's on a journey and meets some unfriendly travelers who learn of his identity and call him 'bastard' to assert their supposed superiority.