Well, it depends on who is flirting with her in the fanfiction. If it's someone she doesn't like, she'll probably be dismissive and give cold responses. However, if it's someone she has a bit of an interest in, she could be coy and a bit shy while still maintaining her confident facade.
Percy could be one. He might flirt with her in a more playful, inside - joke kind of way. Like he'll make references to their shared adventures and say something like 'You know, you were really cute when you were fighting that monster back there, Annabeth. And you're even cuter now.' It's a light - hearted and affectionate form of flirting that's based on their deep bond.
One scenario could be in a modern high school AU. A new student, who is a bit of a bad boy, starts flirting with Annabeth in the hallway. He compliments her intelligence in a rather cheeky way, like 'Hey, smarty - pants, you're too brilliant for this place. How about you and I skip class?'. Annabeth, of course, is both flattered and annoyed at the same time.
Maybe an antagonist who wants to disrupt the group dynamic. If Annabeth is a key part of a group, punching her could cause chaos and confusion among her friends. It could also be someone who has been influenced by dark magic or some sort of mind - control in the fanfiction. This would make them act against their normal nature and punch Annabeth without really having a personal grudge.
Her family could be a great help. If they are aware of the situation, they can step in. Maybe her parents can talk to the school authorities or even directly to the bullies' parents. Family support is always crucial in such situations.
One example could be a situation where Annabeth sees Percy getting a lot of attention from a new girl in their group. She might start acting cold towards Percy, which is a classic sign of jealousy in fanfiction. In such a story, the author might explore her insecurities about her relationship with Percy.
Maybe it's a human who has somehow discovered the world of demigods and is afraid of their existence. They shoot Annabeth out of fear and misunderstanding. Another option is that it's a rogue demigod who has gone bad and is working against their own kind. This character could have a personal grudge against Annabeth specifically, perhaps because of her intelligence and leadership skills which they envy.
It could be an enemy from the underworld. Maybe a powerful demon or a vengeful spirit that has a grudge against her. They might use their dark powers to harm her, like sending a bolt of dark energy her way.
There's a chance it could be a new character introduced in the fanfiction. Maybe someone who has a mysterious past and enters Annabeth's life briefly but leaves a big impact, resulting in her pregnancy.