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.
Hermione could appeal to Snape's sense of loyalty. She could remind him of his loyalty to Dumbledore and how she too is loyal to the same cause - the betterment of the wizarding world. By framing her help in the context of a shared goal, such as protecting Hogwarts or promoting peace among wizards, Snape may be more inclined to let her help. Also, she could share some of her own vulnerabilities and experiences, making herself more relatable to Snape and breaking down the walls of his initial resistance.
She might use her knowledge as a way to gain his respect. If she can prove that she is well - versed in complex magic, especially in areas that Snape values like Dark Arts defenses or advanced potions making, he may be more willing to accept her help. She could, for instance, offer a new approach to a difficult potion recipe that Snape has been struggling with. This would show him that she has something valuable to offer and is not just there to meddle in his affairs.
Well, Snape could use his strict teaching style in a positive way. He might push Hermione out of her comfort zone. By giving her tough tasks related to her fears, like facing a certain dark magic creature. And as she accomplishes these tasks, her fears gradually disappear. Also, his presence alone, with his reputation for being powerful, could make Hermione feel safer and more able to confront her fears.
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.
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.
Their relationship could start off very tentatively. Snape would be his usual gruff self, but Hermione would notice his small acts of help. As time goes on, she might start to trust him more and they could develop a sort of mentor - mentee relationship, with Snape passing on some of his more esoteric knowledge of magic to Hermione.
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.
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.
Well, in some fanfictions, Hermione might use her vast knowledge of magic. She could research rare spells or potions to heal whatever ails Snape. Maybe she discovers a way to reverse the effects of a dark curse that was placed on him. For example, if the curse is sapping his strength, she might find an ancient counter - curse in the library that requires precise incantations and rare ingredients. She would then gather those ingredients, perform the spell, and slowly help Snape regain his health and vitality.
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.
Snape would probably be quite surprised at first. Hermione is known for her academic prowess and self - sufficiency. His reaction could range from being dismissive, thinking she should be able to solve her own problems, to being intrigued by the nature of her request. If it's something related to the dark arts or a mystery in the wizarding world, his interest might be piqued, but he'd still be guarded in his response. He might even try to use her request as a way to gain some advantage or information for himself.