It could signify trust. Annabeth trusts Percy enough to be in such an intimate physical position. It shows that she feels safe with him.
It could be a symbol of Percy's growth and maturity. His physical development might represent his journey from being a young demigod to a more confident and capable one.
It could symbolize a strong bond between them. Dumbledore is like a father figure to Harry, and this physical closeness shows the level of trust and affection Harry has for Dumbledore and vice versa.
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.
He could be a bit surprised at first, but then a warm feeling would spread through him. He might gently stroke her hair or cover her with a jacket if she seemed cold, all while having a soft smile on his face.
Percy might feel a deep sense of contentment. He and Annabeth have a special bond, and having her trust him enough to sleep in his lap could make him feel warm and happy inside. He may also feel a bit nervous, not wanting to move and disturb her peaceful slumber.
One possible scenario could be that they are on a long journey and Annabeth is exhausted. Percy offers his lap as a place for her to rest, showing his care for her. Another might be after a difficult battle where Annabeth is drained, and Percy's lap becomes a safe and comfortable place for her to recover.
One challenge could be office gossip. People might start talking if they notice Percy and Annabeth getting too close. This could create a difficult work environment for both of them. Another challenge is the power dynamic. Percy has to be careful not to abuse his power as a boss when it comes to his relationship with Annabeth.
It could be set in a world where they are undercover in a place like a cabaret for a quest. Annabeth uses the lap dance as a distraction to get information they need. But during the act, Percy is so flustered that he almost blows their cover, and they have to deal with the aftermath of that close call.
Maybe she had a long day of training or adventuring. She could have been mentally and physically exhausted from dealing with various challenges in their world, like fighting monsters or solving complex puzzles.
It could be a sign of closeness. Wearing his hoodie might show that Annabeth feels a special connection to Percy, like she wants to be close to him in a very personal way.