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.
Maybe Harry discovers something about his past, perhaps related to his parents, that only Snape can help with. Snape was closely associated with Harry's parents. He might know some secrets or have some information that could help Harry understand a family mystery or some hidden power within him.
He could also be a bit sarcastic in his response. Snape has a sharp tongue, so he might say something like 'Miss Granger, what makes you think I have the time or inclination to help you?' But as Hermione explains her situation, he might gradually soften, realizing the seriousness of the matter. Maybe he starts to see a bit of himself in Hermione's thirst for knowledge.
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.
Perhaps Hermione is injured in a fight against dark wizards while on a secret mission. Snape discovers her and, despite their past differences, decides to help. He uses his skills in dark magic knowledge to counteract the dark spells that have injured her. This could lead to a begrudging respect between them as they work together to heal her and keep the situation a secret from other Hogwarts students and staff.
The conflict could be Snape's pride. He is a proud man and asking for help, especially from a student like Hermione, goes against his nature. So he constantly battles his own ego while Hermione has to deal with his snarky remarks.
Maybe Snape has a difficult spell he can't seem to master, and as much as he hates to admit it, Hermione's academic prowess makes her the best person to ask for help. He approaches her in a very snarky way, but she sees through his facade and decides to assist him. This could lead to an interesting dynamic where they start to respect each other more.
Snape would probably be very cold and indifferent at first. He might even think it's some sort of Gryffindor ploy to get his attention or make him look bad. He has always had a low opinion of Hermione's house, so his initial reaction would be guarded.
One possible reason could be a misunderstanding. Maybe Hermione thought Snape was involved in some dark plot against her friends. In the wizarding world, there are often secrets and deceptions, and if she got some false information, it could lead her to take such drastic action.
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.