Perhaps he's feeling guilty about something that happened during one of his quests. Maybe he made a decision that had negative consequences for someone he cares about, and that's weighing on him. It could also be that he's just had a really tough encounter with a powerful enemy and is letting out the pent - up stress and fear through crying.
She might immediately go to comfort him. Annabeth is a strong character, but she also has a soft side for Percy. She could hold him and let him cry it out without asking too many questions at first.
One possible fanfiction could be 'The Unseen Pain'. In this story, Percy might be crying due to the overwhelming pressure of his demigod duties. Nico stumbles upon him in a secluded corner of Camp Half - Blood. Nico, initially hesitant, decides to approach Percy and offer his silent support.
Maybe he's missing his family. He's been away from them for a long time on the Argo 2, and homesickness could hit hard and make him cry.
Another possibility is that she had an argument with a friend. It might not even be Ron. It could be with Harry or another Gryffindor. She's very sensitive about her friendships and any conflict could make her cry.
A common reason could be the pressure of her role. As a guardian, she has a lot on her shoulders. If she makes a mistake, it could have serious consequences. So, the stress might make her break down and cry.
It could be that he's lost someone important. Percy has had many friends and family in his life, and if he loses one of them, it would be a huge blow.
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.
One possible reason could be unrequited love. In the fanfic, Chansung might be in love with someone who doesn't feel the same way. This could lead to a deep sense of sadness and cause him to cry. For example, he might have confessed his feelings but been rejected, leaving him heartbroken.
Perhaps she's being bullied. In the wizarding world, there could be other students who pick on her because she's a 'Mudblood'. This could be really distressing for Hermione, and when Snape finds her crying, he might be conflicted between his general dislike for Gryffindors and his sense of justice.
One possible reason could be Percy's recklessness. He often jumps into dangerous situations without fully thinking them through, and Annabeth, being the more strategic one, might get frustrated and yell at him for endangering himself and others.