Her intelligence allows her to understand complex magical concepts faster. In class, she might answer questions before Snape even finishes asking, showing that she doesn't need his full explanation. This would be really irritating for Snape as he likes to be in control of the learning process. Also, she can use her knowledge to find loopholes in Snape's rules. If he sets a task, Hermione might find a more efficient way to do it based on her extensive reading, which would make Snape feel like she's undermining his authority.
Hermione's intelligence gives her the confidence to question Snape's methods. She can reference advanced magical texts that Snape may not have expected her to read. For instance, when Snape is teaching a spell, Hermione could mention a different way of casting it from an ancient spellbook. This not only shows that she's smarter but also that she doesn't fully trust his teaching, which would be extremely annoying to Snape.
Hermione's intelligence means she can quickly spot flaws in Snape's statements. So, she can correct him, which annoys him. For example, if Snape makes a small error in a potion ingredient amount, Hermione will point it out right away.
Well, perhaps Hermione starts debating with Snape about magical theories during class. She might bring up new research or different interpretations that go against what Snape has been teaching for years. Like, when Snape is talking about the properties of a certain magical plant, Hermione could say 'But Professor, I've read in an old wizarding journal that the plant has additional properties that we're not considering.' This would surely annoy Snape as he's not used to being challenged in such a way.
Hermione's intelligence allows her to understand Snape's complex nature. She can analyze his actions and words more deeply than others. For instance, when Snape is being particularly harsh in class, Hermione realizes it's often because he's under a lot of pressure or has high standards. So she can comfort him by saying she understands his intentions. Her knowledge also helps her to find solutions to problems that are bothering Snape. If there's an issue with a potion or a magical object, Hermione can use her vast knowledge to fix it, which in turn comforts Snape as it eases his stress.
Snape's brooding and mysterious personality might work in Hermione's favor. His intimidating presence alone could deter some of her enemies. People in the wizarding world might be afraid to cross him, so they would think twice before harming Hermione if they knew Snape was protecting her.
She might be initially defensive. Hermione is a proud and independent character. So, when Snape finds out about her arm, she could be worried that he'll use it against her in some way, like taking away house points or using it as an excuse to give her extra work.
One possible reason could be the stress from her studies. Hermione is always under a great deal of pressure to excel academically. Maybe she reached a breaking point due to the overwhelming workload. Another reason might be some kind of conflict with her friends. For example, if she had a big fight with Harry or Ron over something serious, it could leave her feeling broken.
A common theme could be redemption. Snape, who has a complex and often - dark past, might find in Hermione a chance to redeem himself in some way. Maybe he saves her from a danger in the dungeon, which starts his journey towards a more positive path. Another theme is the exploration of hidden knowledge. The dungeon is full of secrets, and both Snape and Hermione are curious minds. They could uncover something that changes their understanding of Hogwarts or the wizarding world.
Snape might be suspicious at first. He might think it's some sort of trick or part of a bigger plot. However, as he observes Hermione more closely and realizes the genuineness of her situation, his attitude could gradually change to one of reluctant assistance. He might even offer some words of wisdom, albeit in his typical sarcastic way, to help Hermione regain her strength.
He could use his position of authority. Snape might confront the abuser directly. Since he's a teacher, he has the power to intimidate and put a stop to any immediate harm. He could also give Hermione a safe place to stay, like his own quarters (if it's within the rules of the fanfiction world).
It could be a failed experiment in potions class. Hermione is always eager to try new things in potions, and perhaps this time it backfired, causing her some harm. And Snape would be the one to find her because it's his class.
You could write about their interactions in a time - travel scenario. Maybe Hermione travels back in time and has to work with a younger Snape to solve a mystery that could change the future of Hogwarts. This allows for creative exploration of their personalities and how they might react to each other in an unusual situation, while still maintaining the integrity of their characters.