It could have a huge impact. For example, the child might prevent some of the major tragedies that happened in the original story. Maybe they could stop the Red Wedding from occurring, which would completely change the power dynamics in the Seven Kingdoms.
In many fanfictions, their meeting is a catalyst for change in the fight against the White Walkers. Jon Snow has first - hand knowledge of the threat, and Daenerys has the resources to combat it. Together, they are able to mobilize more forces and come up with better battle plans. Their union, whether it's a political or a romantic one, gives hope to the people of Westeros that they can be saved from the undead horde, which in turn shapes the rest of the fanfiction story.
They could be allies. Sirius, being someone who values justice and freedom, would likely be drawn to Daenerys's cause of freeing the slaves. He might see her as a powerful leader with a good heart. Daenerys, on the other hand, would probably respect Sirius's combat skills and his loyalty. They could strategize together, with Sirius providing his insights on how to deal with enemies and build strong alliances.
If Jon Snow hatches a dragon in the fanfic, it would completely disrupt the power balance in the Game of Thrones world. Dragons are extremely powerful creatures, and having one on Jon's side would make him a formidable force. He could use it to quickly travel across Westeros, intimidate his enemies, and gain more support. This would likely cause other rulers to either try to ally with him or plot against him more vehemently. The Night King and the White Walkers would also have a new and dangerous adversary, as a dragon could be a very effective weapon against them.
One interesting fanfiction could be where Jon and Daenerys form an alliance much earlier and together they face the White Walkers. They combine their forces and knowledge, with Jon teaching Dany about the North and its ways, and Dany sharing her dragon - riding skills with Jon. This could lead to a more united front against the common enemy.
I've read one where Jon actually has a vision of Daenerys before they meet. He sees her as a key to saving Westeros from the impending doom of the White Walkers. When they finally meet, there's an instant connection. It's a really well - written fanfiction that delves deep into their characters and the complex world of 'Game of Thrones'.
In some fanfictions, the baby could become a unifying force. Since Jon has a claim to the North and Daenerys had a claim to the Iron Throne, the baby might be seen as a rightful heir to both. It could bring an end to the long - standing feuds between the different regions of Westeros.
One scenario could be that Jon and Daenerys meet earlier in their journeys. Maybe Jon, while on an expedition beyond the Wall, stumbles upon Daenerys and her dragons in Essos. They form an immediate bond over their shared sense of justice and leadership. Daenerys is intrigued by Jon's loyalty to his people in the North, and Jon is in awe of her power and determination.
One interesting plotline could be their joint quest to defeat the Night King. They combine their forces, both the Unsullied and Dothraki with the Northmen. Their relationship also evolves during this time, from initial distrust to a deep mutual respect and perhaps something more romantic. They learn from each other's leadership styles and together become a formidable force against the undead threat.
Another fanfiction concept could be a 'what if' scenario. What if Jon was the one who found Daenerys when she first landed in Westeros? This would put Jon in a position of power and knowledge about this new queen. He could act as her guide through the complex web of Westerosi politics. Their relationship might develop more organically as he helps her navigate the land he knows so well, and perhaps they would fall in love more gradually, away from the prying eyes of the other lords and ladies.
One great fanfic is 'Snow and Dragons'. It beautifully weaves the relationship between Daenerys and Jon. It starts with their first encounter and how they gradually come to respect and then fall in love with each other. The author does a wonderful job of capturing their personalities as seen in the show.