It could be part of a larger plan to change the wizarding world's view of werewolves. By having a child with Remus, Hermione could be trying to show that werewolves are not to be feared or shunned. She might also be motivated by a personal growth aspect, wanting to step out of her comfort zone and do something radical to make a difference. Also, she may have a strong maternal instinct that she wants to direct towards Remus' situation.
One reason could be her sense of justice and compassion. Remus is a werewolf and faces discrimination, and Hermione might see having a child with him as a way to give him a family and some normalcy in his life.
In fanfiction, Sirius and Remus could fight over Hermione because of their different relationships with Harry. Since Hermione is Harry's close friend, Sirius might think that he should have more influence over Hermione to help Harry better, as he is Harry's godfather. On the other hand, Remus, being Harry's former teacher and friend, also feels a sense of responsibility towards Hermione because of her connection to Harry. This clash of responsibilities and their own personal feelings for Hermione can spark a fight.
Maybe in this fanfiction, Remus and Hermione have shared some deep intellectual conversations that Tonks couldn't engage in as much. Hermione's knowledge of magic, both theoretical and practical, might have drawn Remus to her. Also, Hermione's determination and loyalty could be factors. She is fiercely loyal to her friends and would go to great lengths to protect them, which Remus might value highly.
In the fanfic, it might be due to some sort of magic gone wrong. There could be a spell that made her act out of character. It's not unheard of in the wizarding world for spells to have unexpected effects on a person's behavior and relationships.
Perhaps in this fanfic, the Weasleys have some actions that conflict with Harry and Hermione's values. For example, if the Weasleys were to make a deal with a dark wizard for the sake of their family's safety and Harry and Hermione found out. This could lead to a rift and hatred. Also, it could be related to a love triangle situation in a very convoluted way. Maybe one of the Weasleys unknowingly gets in the way of Harry and Hermione's relationship, or vice versa, which causes animosity.
It could be that they lied to Harry about something really significant. In the wizarding world, trust is built on truth - telling. If they told a big lie, like about the true nature of a magical object that was putting Harry in danger, he would be very angry. And if their actions led to the death of someone Harry cared about. For instance, if they made a wrong move during a fight that caused an ally to die, Harry would find it very hard to forgive them.
One possible reason is that the fanfic writer wants to create conflict. By making Hermione not believe Harry, it adds tension to the story.
In fanfics, the relationships between characters can be explored in new ways. If the fanfic writer decides to show a rift between Hermione and Harry for the sake of drama, they might create situations where Hermione doesn't believe Harry. This could be due to a series of miscommunications or actions that Harry takes which seem out of character to Hermione.
It could be due to a misunderstanding. In the fanfic, Remus might say or do something that Harry misinterprets as being against him. For example, Remus could be trying to protect Harry but his actions come across as controlling, which would make Harry dislike him and potentially hate him.
It would probably shock most of the other characters. For instance, Ginny might be very surprised and might not understand Hermione's motives at first. Hermione's offer could create a divide between those who support her and those who think she's made a very strange and inappropriate decision. This could lead to a lot of tension and conflict within the wizarding community depicted in the fanfic.
One possible reason could be social class differences. Hermione's parents are Muggles and might view Ron's large, wizarding family with some prejudice. They may see the Weasleys as a bit chaotic and irresponsible because of their financial situation.