Well, it could be that he's feeling the weight of his destiny. Harry has so much on his shoulders, like fighting Voldemort. Maybe he just had a particularly tough encounter with the Dark Arts or lost someone dear in the fight against the dark side. Another reason might be his feeling of isolation. Despite having friends, there are times when he feels alone in his battle.
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.
Often, Harry will offer her a handkerchief or a tissue if there is one available. He might also put his arm around her in a comforting gesture. In 'The Common Room Crisis' fanfic, he does just that and then stays with her, listening intently as she pours out her feelings. He is patient and understanding, which is in line with his character in the original books. He tries to make her laugh or at least smile by sharing a joke or a funny memory. This shows his deep friendship and care for Hermione.
Perhaps he's feeling isolated. Even though he has friends, he often has to face his battles alone in some ways. Ginny finding him crying could be a moment of vulnerability that she witnesses, and it might be related to his feeling of being alone in the fight against the dark forces.
Harry might immediately rush to her side and comfort her. He could put his arm around her and ask her what's wrong in a gentle voice.
One great fanfiction could be 'The Unspoken Pain' where Harry stumbles upon Hermione crying in the library. It delves into the stress she's under from all the studying and the looming threats in the wizarding world. Another is 'A Moment of Weakness' which shows a more personal side of Hermione, perhaps feeling left out by her friends, and Harry is there to comfort her.
Another great one is 'When Draco Saw Harry Cry'. This fanfiction focuses on the moment of discovery and how it changes the dynamic between them. It has some really heart - wrenching moments as Harry is dealing with his own issues and Draco has to decide how to respond.
She probably comforts him right away. Hermione is a very empathetic person. She might give him a big hug and let him cry on her shoulder.
Well, it could be due to the stress of their relationship. Harry had a lot on his plate, being the hero and all. Ginny was also a strong - willed witch with her own ideas. They might have had an argument about their future together. For example, Harry might be worried about Ginny's safety if they stayed together, given his dangerous life. Ginny, on the other hand, could be frustrated that Harry was trying to push her away for her own good. Another possibility is that they could be crying over the state of the wizarding world. After all the destruction caused by the war, seeing their beloved Hogwarts in ruins or the wizarding community divided could be heart - breaking for them.
It could be that she's homesick. Even though she loves being at Hogwarts, sometimes she might miss her muggle family a great deal. Also, she might be crying because she's being bullied. There could be some students who are jealous of her intelligence and are giving her a hard time. In the fanfiction, Sirius finding her crying might lead to him becoming a sort of mentor or protector for her, helping her deal with whatever is making her sad.
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.