Perhaps there's a new dark magic item or spell that has corrupted her mind. In the wizarding world, there are many powerful and dangerous things. If Hermione were to come into contact with one that warped her sense of right and wrong, she might turn against Dumbledore. Another idea is that she's trying to protect someone she loves from Dumbledore, believing that he poses a threat to them.
She could be misinformed. For instance, if someone she trusts tells her false things about Dumbledore's past or his intentions. It's also possible that she discovers something about Dumbledore that she believes is unforgivable, like a secret alliance with dark wizards.
It could be a matter of ideological differences. Dumbledore has his own way of handling things in the wizarding world, and Hermione might develop a different view as she grows older and more experienced. If she feels that Dumbledore's methods are causing more harm than good, in a very extreme sense, she could consider murder as a solution. However, this would be a huge departure from her normal character, which is what makes it such an interesting fanfiction concept.
This could be a very dark and interesting fanfiction concept. Maybe Hermione has been deceived by some false information about Dumbledore, thinking he has committed unforgivable acts. Or perhaps she's under some sort of dark magic influence that makes her believe murdering him is the right thing to do. It could also be a plot device to explore the complex character of Hermione and how far she can be pushed from her normal moral compass.
The professors may desire Hermione fanfiction because it often delves deeper into the 'Harry Potter' universe. Hermione has so many relationships and storylines that can be explored further in fan - made stories. Also, fanfiction can provide unique perspectives on her character growth, which might be interesting for professors who study literature or character development.
One possible reason could be that she broke a school rule in the fanfiction. Maybe she was caught using magic outside of class in a way that was against Hogwarts' regulations.
Dumbledore might be motivated by his over - arching plan to defeat Voldemort. He could see Hermione as a sacrifice that needs to be made. Maybe he believes that by betraying her, he can gain some crucial information from the enemy that will ultimately lead to Voldemort's downfall.
One possible reason could be a miscommunication. Maybe some evidence is misinterpreted, and Dumbledore thinks Hermione has broken a major rule. For example, if he thought she was involved in some dangerous magic experiments without permission. But this is a hasty conclusion for him to make.
One possible plot could be that Hermione discovers a hidden family secret that reveals she is Dumbledore's sister. This leads to her having access to new knowledge and powers that Dumbledore left for her, which she uses to fight a new dark force in the wizarding world.
In a more extreme case in the fanfiction, Dumbledore might assign Hermione to a special project that is both difficult and dangerous as a form of punishment. For example, he could send her to study a group of cursed magical creatures in the Forbidden Forest. This would be a punishment that not only tests her skills but also her courage.
In some fanfictions, Dumbledore might get punished for his past mistakes. For example, his actions regarding Tom Riddle when he was at Hogwarts could be seen as negligent in hindsight. He might have missed signs that could have prevented Voldemort's rise, and the wizarding community in the fanfiction could hold him accountable for that.
One possible reason could be that she discovers some of Dumbledore's past actions that seem morally ambiguous. For example, his handling of the situation with Tom Riddle when he was at Hogwarts. Maybe she finds out some details that make her question his judgment.
It's also possible that Dumbledore has been misinformed or is under the influence of a dark spell himself. In the wizarding world, there are many forces at play. He could have received false information that made him think this was the best course of action for Harry. Another aspect could be that he is trying to teach Harry a lesson in a very extreme way. Maybe Harry has been too reckless, and Dumbledore wants to make him understand the seriousness of the situation.