They could have adventures where they explore ancient magical places. Maybe they find a long - lost magical artifact that has the power to change the fate of Camelot. Merlin, with his knowledge, and Aithusa, with its dragon abilities, would be a formidable team in such quests.
Their relationship could develop in a rather complex way. Merlin, having his own destiny and responsibilities, still makes time for Aithusa. He might take Aithusa to hidden places where it can learn and grow safely. Aithusa, in turn, looks up to Merlin. It may show its loyalty by protecting Merlin when he is in danger. They form a bond that is not only based on blood but also on mutual understanding and a shared sense of magic.
I'm not sure of a specific fanfiction for this scenario off the top of my head. But you could try searching on Archive of Our Own. There are likely some great stories out there where Gwaine witnesses Merlin hatching Aithusa.
Some Merlin - focused fan communities or forums might also be a good source. Members often share and recommend their favorite fanfictions. You could join a Merlin fan forum and ask for recommendations for Merlin Aithusa fanfictions specifically.
One common theme is friendship. Since Merlin has a special connection with Aithusa, fanfics often explore how their friendship grows and evolves. For example, they might start off as cautious allies and become true friends over time.
In some fanfictions, Aithusa could play a role in Merlin's discovery of more of his powers. Since dragons are so closely tied to magic in the Merlin universe, Aithusa might show Merlin a new aspect of magic that he was previously unaware of. This could be through some sort of magical vision or by guiding Merlin to a place full of ancient magic.
One popular 'Merlin raises Aithusa' fanfiction might be on Archive of Our Own. You can search for tags like 'Merlin', 'Aithusa' and 'raising' to find relevant stories. Some fanfictions in this theme could explore Merlin's magical bond with Aithusa and how he teaches the young dragon about the world of magic and Camelot.
First, you need to do some research on the Merlin universe, especially about Merlin's character and his relationship with magic and dragons. Then, think about how Merlin would handle raising a young dragon like Aithusa. You could start with a basic plot, like Aithusa being found by Merlin in a dangerous situation and Merlin deciding to take care of it.
There is a fanfic called 'Aithusa's Growth and Merlin's Role'. This one delves deep into how Aithusa develops over time and Merlin's influence on that growth. It shows a very different side of their connection compared to what we see in the original series. The author does an amazing job of creating new scenarios and emotions within the world of Merlin and Aithusa.