Molly could have made some inappropriate comments about Hermione's blood status. Even though Hermione is a Muggle - born, she has proven herself time and time again in the wizarding world. If Molly makes a snide remark about her blood, Hermione would surely tell her off for her prejudice.
It may be because of Molly's interference in Hermione's friendship with Ginny. Maybe Molly doesn't approve of their friendship for some reason. But Hermione values Ginny's friendship and will not tolerate Molly's meddling. So she tells Molly off to protect her friendship with Ginny.
One possible plot could be that Hermione tells off Molly for being overprotective of Ron. Maybe Molly is constantly interfering in Ron and Hermione's relationship, and Hermione finally has enough and gives her a piece of her mind about respecting their privacy and decisions as adults.
One possible reason could be jealousy. Maybe Molly is jealous of Hermione's intelligence and the special bond she has with Harry and Ron. Another reason might be that she feels Hermione is a bad influence on her sons in some way. For example, she might think Hermione's Muggle - born status could bring trouble to the family in the wizarding world which is full of pure - blood supremacist ideas.
One possible reason could be that she might think Harry is putting himself in unnecessary danger, like when he goes off on his own to face Voldemort without a proper plan. Molly is very protective, and she may feel frustrated with Harry's recklessness.
One possible reason could be a misunderstanding. Maybe Molly does something that Harry misinterprets as a threat to his friends or his mission. For example, if Molly tries to keep Harry from a dangerous but necessary task, Harry might see it as her overstepping and feel the need to 'punish' in the context of the fanfiction, which could just be a strong rebuke or some form of isolation within the story's plot.
One reason could be Hermione's excessive worrying. She often frets over small details and Harry might get tired of it and tell her off. For example, if she keeps nagging about the proper way to perform a spell when they are in a hurry during a dangerous situation.
One possible reason could be a disagreement over how Molly treats Harry. Maybe Molly is being overprotective of Harry in a way that Hermione thinks is hindering Harry's growth as a wizard. For example, if Molly tries to stop Harry from taking certain risks that Hermione believes are necessary for his development.
One reason could be that the fanfiction changes her personality traits. In the original, she's kind - hearted but in fanfic, she might be made mean. Harry, who knows the real Molly, won't like this new version.
One possible reason could be that Molly might blame Harry for putting her family in danger. Since he is often at the center of Voldemort - related troubles, her family gets involved in protecting him. For example, during the Battle of Hogwarts, her family was right in the middle of the fight mainly because of Harry. Another reason could be that she blames Harry for Ron's decisions at times. Ron follows Harry on his adventures and sometimes it puts Ron in harm's way, which Molly as a mother may not like.
It could be that Ron is having a bad day or going through some personal issues. And instead of dealing with them in a healthy way, he takes it out on Hermione. For example, if he's feeling insecure about his place in the group, he might lash out at her.
Well, Hermione has a unique charm. In fanfiction, Bill could be attracted to her because of her strong sense of justice. She is always fighting for what's right, and Bill, who has seen a lot in the wizarding world, might admire that. Also, her loyalty to her friends is something that could draw Bill to her. He might see in her a kindred spirit in terms of being there for those he cares about.