One idea could be that they meet in a magical library that exists between their two worlds. Annabeth is seeking knowledge about a new type of magic she's heard of, and Hermione is researching ancient civilizations. They start sharing their knowledge and end up working together on a project to save both their worlds from a common threat.
Maybe they meet at a wizard - demigod conference. At first, they're a bit skeptical of each other, but as they start talking, they realize they have a lot in common. For example, they both have a thirst for knowledge and are very brave. They then team up to solve a mystery that has been plaguing both the wizarding world and the demigod world, like a series of strange disappearances.
They could start off by competing in a friendly competition within their camp. This would show their competitive sides and also how they interact when there's a bit of rivalry. Then, they might realize they make a better team when they accidentally uncover a mystery related to the camp's history.
A plot could be set in a post - apocalyptic world where Percy and Annabeth are on opposite sides at first due to some misunderstanding. As the story progresses, they realize they need to work together to face the real threats in this dark new world.
One plot idea could be that Percy is so caught up in his own heroic deeds and adventures that he completely misses all of Annabeth's subtle hints of her feelings for him. Meanwhile, Annabeth tries different ways to get his attention, like helping him with his studies in a very flirtatious manner, but Percy just thinks she's being a good friend. Another could be Percy getting into a situation where he has to protect Annabeth without realizing his true feelings for her are deeper than friendship, and throughout the fanfic, his obliviousness slowly starts to fade as he gets closer to her in the face of danger.
In a fanfiction about their bickering, a plot could be centered around a magical item. Let's say they find a new weapon or artifact. Percy wants to use it right away to test its power, but Annabeth is worried about the potential consequences and unknown abilities of the item. This leads to their bickering. Additionally, they could bicker about their relationships with other characters. For instance, Percy is friendly with a new demigod who Annabeth is a bit suspicious of, and this causes tension and arguments between them.
One idea could be that Percy and Annabeth are on a pirate ship, and they discover a hidden treasure map. They have to battle other pirates and sea monsters to reach the treasure. Another could be that they are pirates who are trying to free a group of captured demigods. A third idea is that they are pirates on a quest to find a magical artifact that can control the seas.
One idea could be a post - quest moment where Percy and Annabeth are sitting by the Camp Half - Blood lake at sunset. They start sharing their deepest fears and hopes, and as they get closer emotionally, they hold hands and share a gentle kiss. It's a very tender and intimate moment that shows their deep connection.
Well, a great plot could be that they find a mysterious object under their bed. It turns out to be a key to an ancient demigod treasure. So they start researching in bed, looking through old scrolls and books, trying to figure out where this treasure might be. And during this process, they also share some sweet and tender moments, reminiscing about their past adventures. Also, they could be in bed during a stormy night at Camp Half - Blood and tell each other spooky stories about the monsters they've faced, which leads to them coming up with new strategies for future battles.
One idea could be a modern-day high school AU. Percy could be the new kid at school, a bit of a loner and a troublemaker. Annabeth, on the other hand, is the top student and part of the student council. They get paired up for a project and initially clash because of their different personalities. But as they work together, they discover they have more in common than they thought, like their love for the ocean (Percy's always been drawn to it, and Annabeth has a fascination with ancient sea civilizations).
Maybe they are rivals in a quest given by the gods. Each wants to prove themselves as the better hero. Percy uses his combat skills and connection to the water, while Annabeth uses her strategic mind and knowledge of ancient places. This could lead to some intense moments where they have to decide whether to cooperate or keep competing.
Another idea is that Aphrodite, out of jealousy because Annabeth is so smart and independent, decides to put Annabeth through a series of trials related to love. Annabeth has to face her own insecurities about love and relationships as she overcomes these trials. In the end, she emerges stronger and more confident, and Aphrodite has a newfound respect for her.