It could be that she's homesick. Despite all the adventures, sometimes the longing for home can be overwhelming. Percy would comfort her by sharing his own feelings of missing his mother and how they can make new homes with their friends. He might hold her hand and tell her that their friends are like family now.
Annabeth is known for her intelligence and strategic thinking. She is a strong and independent character. Percy, on the other hand, is brave and has a strong connection to the water due to his father being Poseidon. In the fanfic, when Annabeth sits in Percy's lap, it might be a way to show their unique relationship which has developed over the course of their many shared experiences in the Percy Jackson universe.
One common theme is redemption. Hera has made some mistakes in the Percy Jackson series, and in these fanfics, Percy might be the one to help her see a path to redemption. For example, Percy could offer forgiveness and understanding that leads Hera to re - evaluate her actions.
Another idea is that they are rivals in winning the favor of a powerful but mischievous god. The god sets tasks for them, and whoever completes the most tasks successfully will get a special reward. Percy's courage and Annabeth's wisdom are put to the test. This rivalry might also test their friendship as they are tempted to use underhanded tactics against each other but ultimately realize the value of their bond.
One great fanfic could be about Percy and Annabeth competing in a series of challenges set by the gods. Each challenge tests their skills, knowledge, and leadership. Their rivalry heats up as they both strive to outdo the other. For example, in a maze - like challenge, Percy uses his water - based powers to find shortcuts while Annabeth uses her strategic mind to map out the best routes.
It could be common. Considering how important Annabeth is to Percy in their story, he might be eager to explore more about her through fan - written works. However, it also depends on Percy's personality. If he is the type who likes to stick to the original story only, then it might not be so common. But if he has an open - minded attitude towards different interpretations of their relationship, then it's likely to be common for him to like Annabeth fanfic.
One idea could be that Percy gets sent on a quest in a different realm while Annabeth is stuck in Camp Half - Blood dealing with internal camp issues. This separation could lead to both of them growing individually. Percy might learn new powers or make new allies in the other realm, and Annabeth could strengthen her leadership at the camp.
One great 'Percy and Annabeth fanfic' is 'The New Adventures of Percy and Annabeth'. It takes their relationship to new places, like exploring different mythologies together. Another one is 'Percy and Annabeth: College Years' which shows how they adapt to a more normal life while still having their demigod challenges. And 'Annabeth's Quest with Percy' is also good as it focuses on Annabeth leading a quest with Percy by her side.
I'm not sure if there is a specific 'Percy and Annabeth bikini fanfic' available. There are tons of fanfics out there, but it would take some searching through fanfic platforms like Archive of Our Own or FanFiction.net to find one with that exact theme.