It's hard to say who would be the clear winner. Hermione is intelligent and quick - witted, but Snape has years of experience and a great deal of magical knowledge. If it's a battle of words, Hermione might hold her own for a while, but Snape could use his authority to shut her down.
In a way, there might not be a traditional 'winner'. If Hermione confronts Snape over an issue of justice, she could make him see her point of view, which would be a victory of sorts for her. However, Snape could also manipulate the situation to make it seem like he still has the upper hand, even if he's been forced to consider her stance.
One possible scenario could be that Hermione confronts Snape about his unfair treatment of Gryffindor students. Maybe she has evidence of him being overly harsh and she stands up to him in a classroom setting.
Dumbledore might come to Snape's aid. Dumbledore always had a somewhat complex relationship with Snape, and he would likely step in to prevent things from getting too out of hand. He could use his powerful magic to stop Hermione and then try to sort out the situation calmly.
Well, it often starts with a lot of misunderstandings. Snape has his own preconceived notions about Hermione as a Gryffindor student, and Hermione has her own view of Snape as a strict and often unfair teacher. However, as they live together, small moments start to change their relationship. For example, when Hermione is having trouble with a particularly difficult spell, Snape doesn't scold her but instead patiently guides her. This gradually builds trust between them, and they form a unique bond that is different from the typical student - teacher relationship or even a normal father - daughter relationship.
In the fanfiction, the relationship may start off with Snape being rather cold and distant towards Hermione as his familiar. However, as Hermione proves herself time and again, perhaps by using her unique skills to solve problems that Snape couldn't on his own, he starts to warm up to her. Their relationship then develops into a more complex one, where they have a mutual respect and a sort of unspoken understanding. This could involve Snape sharing some of his own magical knowledge with Hermione in a way that he would never do with other students, and Hermione in turn, helping Snape with his personal and professional matters, such as dealing with the other Hogwarts faculty or protecting the school from hidden threats.
At first, Hermione might be suspicious of Snape's new - found protectiveness. But as time goes on, she starts to see that he has good intentions. Snape, on the other hand, gradually warms up to Hermione more and more, sharing his knowledge and advice with her.
At first, their relationship would likely be tense. Snape is known for his dislike of Gryffindors, and Hermione is a prime example of a Gryffindor student. But being in the close quarters of the supply closet might force them to communicate more directly. They could start to see past their initial prejudices. Maybe Snape would recognize Hermione's intelligence and Hermione would see that Snape has reasons for his cold exterior.
A possible way to develop their relationship is through a magical owl - related research project. Hermione, being the studious one, is eager to participate, and Snape is in charge. As they work on the project, they exchange ideas and arguments. Over time, Hermione starts to respect Snape's vast knowledge, and Snape begins to appreciate Hermione's fresh perspectives. This could transform their relationship from a hostile one to a more amicable one.
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.
In fanfiction, Snape might be drawn to Hermione because of her intelligence. She's extremely bright and her thirst for knowledge could be appealing to Snape, who also values intelligence highly. Her ability to think quickly and solve problems might catch his attention.
One possible secret could be about his past relationship with Lily Potter. Maybe he reveals how much Lily meant to him and the complex feelings he had towards her, which in turn influenced his actions throughout the series. Another secret could be related to certain spells or potions that he has discovered but kept hidden from others in the wizarding world.