A meeting between them in fanfiction could be shown as a diplomatic encounter. Maybe Jon is sent back in time for some reason, and he has to negotiate with Aegon. He would use his wits and charm to make Aegon understand the future implications of his conquests. Aegon, being a shrewd leader, would listen intently and might even change some of his plans based on Jon's insights.
One idea could be that they meet during a political intrigue in King's Landing. Female Jon Snow, with her Stark-like honor, might initially be suspicious of Aegon Targaryen. But as they get to know each other, they find common ground in their desire to bring peace to the Seven Kingdoms. They could form an alliance that eventually turns into a deep friendship or even more, all while facing various threats from other power - hungry lords.
One great 'Aegon the Conqueror Fanfiction' is 'Aegon's Ascent'. It delves deep into his early days and how he came to be the conqueror. The character development is really well - done.
One possible plot could be that female Jon Snow, who has the same noble and somewhat conflicted nature as her male counterpart, meets Aegon. They initially clash due to their different upbringings and loyalties. For example, if Aegon has a claim to the throne and Jon Snow is still tied to the North, their paths cross in a political struggle. But as they get to know each other better, they start to understand and respect each other's viewpoints and might even fall in love, changing the political and personal dynamics of the story.
You can try fanfiction websites like Archive of Our Own (AO3). There are often a plethora of fanfictions with different takes on characters. Just search for 'female Jon Snow Aegon' on the site, and you'll likely find relevant works.
Some dedicated fantasy fanfiction forums also have great 'Aegon the Conqueror Fanfiction'. People there are really into this kind of stuff and often share their own works or recommend others.
You could make an OC who is a skilled warrior from a small, overlooked kingdom. This warrior OC initially sees Aegon as a threat but then comes to respect his leadership and power. The fanfiction could detail their encounters on the battlefield and how their relationship evolves over time. For example, the warrior OC might be captured by Aegon's forces and instead of being killed, is given a chance to join Aegon's cause.
In a fan fiction, they might discover that they are related in a more complex way than they thought. Maybe they find out that there is a hidden prophecy that requires both of their bloodlines to come together to bring peace to Westeros. This could lead to a journey where they travel together, facing various challenges and gradually building a bond of friendship and respect.
Developing relationships is key. If your OC is in the world of Aegon the Conqueror, they will interact with various characters. Show how they react to Aegon, his sisters, and the different lords and ladies. Are they in awe of Aegon? Do they despise him? These relationships will shape your OC. For instance, if your OC falls in love with one of Aegon's soldiers, it will add a whole new dimension to their character and the story.
The plot could involve a power struggle. Aegon wants to reclaim the Iron Throne for the Targaryens. Jon, on the other hand, has his own loyalties. They could start as rivals. But then, when a new threat emerges, like a powerful dark magic user who wants to rule Westeros through chaos. They are forced to put aside their differences. Jon's knowledge of the North and Aegon's Targaryen heritage could be combined to fight this new foe.