One way could be that Percy realizes his mistake right away. He quickly apologizes to Annabeth, explaining that he was just having a bad day or was influenced by something he didn't understand. Annabeth, being in love with him, would probably forgive him after some time.
Another possibility is that Annabeth takes some time away from Percy. During this time, Percy reflects on his actions and makes changes in himself. He becomes a better person and then approaches Annabeth again. Annabeth sees the change in him and they reconcile, with their relationship even stronger than before.
Ron could realize his mistake and sincerely apologize to Hermione. He might give her a heartfelt speech about how much he values her and didn't mean to hurt her.
Another way is through the intervention of their friends. Their mutual friends could bring them together and encourage them to see things from the other's perspective. The friends can share stories about how each of them is really a good person deep down. This might soften their hearts towards each other.
Communication is key. They need to sit down and talk about their feelings. For example, Percy could tell Annabeth that he's only being friendly with the new demigod who seems to like him, but his heart is with Annabeth. Annabeth could then share her insecurities. Another way is through a shared adventure. If they go on a quest together where they have to rely on each other, they'll realize that their relationship is stronger than any jealousy - inducing situation. They could also be given a task by Chiron that forces them to work together and face their jealousy head - on.
She could also talk to the author of the fanfiction. Maybe there was a misunderstanding or the author could clarify some things that are scaring her. For instance, if it's about Percy's character, the author could explain their creative choices.
They could have an honest conversation. Just sit down and talk about their feelings and misunderstandings. This simple act can clear the air and resolve the hatred.
There could be many authors. It might be a new writer in the fanfiction community who wants to explore this rather dark and emotional side of their relationship.
In a fanfiction, it could be that Percy gets influenced by some sort of magic or external force that makes him act coldly towards Annabeth. He starts ignoring her and when she tries to confront him, he says some mean things. Annabeth, who is used to Percy being her support, would be heartbroken and cry.
The author of the fanfiction could rewrite it to better align with the characters' personalities. For example, by making Percy more in - character and the high school setting more unique.
Harry could make it up to Hermione by doing something special for her. Maybe he could find a rare book for her library in the wizarding world or help her with a project she's been struggling with. This would show her that he really cares and wants to make amends.
Maybe Annabeth says something really hurtful about Percy's past, like making fun of his insecurities regarding his father. That could make Percy cry.