A common trope is 'character growth'. In fanfics like 'Clarisse's Evolution', we see Clarisse grow from a brash and sometimes mean character to someone more understanding and empathetic. Percy also often experiences growth in these fanfics. Another is the 'mystery quest' trope. For example, in 'The Quest for the Lost Artifact', Percy and Clarisse go on a quest to find a powerful artifact. It involves a lot of mystery - solving and adventure. And don't forget the 'forbidden friendship' trope in some fanfics where their friendship is frowned upon by others at first.
One great fanfic is 'Clarisse's Redemption'. It shows Clarisse in a new light, dealing with her insecurities while still maintaining her tough exterior. Another is 'Percy and Clarisse: An Unexpected Alliance' which focuses on how they put aside their differences to face a common enemy. And 'The Friendship of Percy and Clarisse' is also a good one, exploring how their relationship could develop into a deep friendship over time.
There's also 'Clarisse's Secret Quest'. In this one, Clarisse goes on a solo quest that Percy later gets involved in. It's a great story that shows Clarisse's determination and strength. The author does an amazing job of keeping the essence of the 'Percy Jackson' world while adding their own unique twists to the plot.
The theme of power and growth is also common. Clarisse is a strong character, and in fanfiction, we often see her growing in power and learning to control it better. Percy, too, is constantly evolving in these stories. Additionally, the theme of forgiveness plays a part. Their initial animosity means that there's often a need for one or both of them to forgive the other for past wrongs. This is explored in many fanfictions, showing how forgiveness can lead to stronger relationships and personal growth.
One common theme is the idea of overcoming differences. Since Percy and Clarisse often clash in the original series, fanfics like to explore how they can get past their initial animosity. Another theme could be growth, both as individuals and in their relationship with each other.