Another possibility is that she has to help Snape with teaching. Maybe there are some students who are particularly difficult to handle, and Hermione's intelligence and experience as a student herself could be useful. She could help Snape come up with better teaching methods or ways to engage the students more effectively.
Perhaps Dumbledore wants Hermione to help Snape in uncovering some dark secrets in the wizarding world. Hermione is known for her research skills and her thirst for knowledge. Together with Snape's deep understanding of the darker aspects of magic, they could work towards revealing and dealing with some hidden threats.
Their relationship could start off with a lot of tension. Snape has always had a bit of a prickly relationship with students, especially those he sees as being too much like Gryffindors. However, as they are forced to collaborate, they might find common ground in their love for knowledge. Snape is a highly skilled wizard in many areas, and Hermione is eager to learn. This shared interest could be the starting point for a more amicable relationship.
She might be required to research in the restricted section of the library. Given her penchant for knowledge and Snape's often - dark and esoteric interests, the restricted section could hold the key to whatever he needs help with.
In a fanfiction, Snape might want to help Hermione when she is wrongly accused of stealing powerful magical artifacts. Snape, knowing her character, decides to use his skills as a Potions master and his knowledge of the castle's secrets to find the real culprit and clear Hermione's name. He could be motivated by his new - found respect for her intelligence and her Muggle - born status which he has come to understand better over time.
A plot could be that Snape is suffering from some sort of post - war trauma that no one else seems to notice. Hermione, being the observant and empathetic person she is, decides to help him. She tries different methods like introducing him to Muggle therapies she has read about, or using calming spells and potions she creates herself. As she helps him, Snape begins to open up about his past, his feelings for Lily, and his experiences during the war, which in turn helps Hermione understand him better.
In some fanfictions, Snape could have his own personal vendetta against Dumbledore. Maybe Dumbledore has wronged Snape in the past, and when he sees Harry as a potential victim of Dumbledore's schemes, he decides to help Harry. Another reason could be that Snape has a new understanding of the prophecy and believes that Dumbledore's way of handling it is wrong, so he steps in to help Harry.
One possible theme could be redemption. Snape, who has a complex past, might see Hermione as a way to redeem himself. Maybe he realizes her intelligence and potential and wants to guide her in a different way than he does with other students.
Hermione could start by showing her genuine intentions. For example, she could offer to help Snape with his potions research without asking for anything in return. Since Snape is passionate about potions, this might pique his interest and make him less resistant.
One possible reason could be related to a difficult potion she's trying to brew. Snape is a potions master, and Hermione might be facing an ingredient or a process that she can't figure out on her own. For example, a rare potion that requires precise handling of dangerous ingredients.
Maybe Snape is involved in some sort of mystery or danger within Hogwarts that has to do with magic spells. Hermione, with her vast knowledge of spells and her in - depth study of the Hogwarts library, could be a great help to him in deciphering old spells or magical texts related to the situation.
One reason could be Snape's sense of duty. As a teacher at Hogwarts, he has a responsibility to look after the students, even if they attack him. So, he would put aside his personal feelings and help Hermione.