Perhaps he realizes that Hermione is often an outcast among her peers in some ways, just like he was. So he wants to protect her from the same kind of loneliness and mistreatment that he endured. In fanfiction, this can be a strong motivation for his protectiveness.
Well, one reason could be that he sees her intelligence and thirst for knowledge, which he respects. He might feel a sort of kinship as he was also a very intelligent student himself when he was younger.
In some fanfictions, Snape might keep an eye on Hermione from afar. He would watch her interactions with other students or teachers to make sure she's not being bullied or manipulated. If he sees someone trying to take advantage of her knowledge or skills, he would step in. Another way is that he could offer her extra help or resources for her studies privately, which shows his protectiveness towards her academic growth as well as her well - being.
Perhaps Snape realizes that Hermione is often a target because of her Muggle - born status. In the wizarding world, there's a lot of prejudice against Muggle - borns. Snape, having experienced his own share of hardships and being an outcast in some ways, might empathize with her situation and thus be protective. Also, her friendship with Harry Potter might make Snape more watchful over her, as Harry is always in the center of many dangerous situations and Hermione is closely associated with him.
Maybe it's related to some past event that hasn't been revealed in the original story. There could be a connection between Snape's family and Hermione's lineage that gives him the right or the need to claim her. For example, if there was a magical debt between their families long ago.
In the fanfiction, it might be shown through Snape secretly watching over Hermione during dangerous situations. For example, in a battle or when she's being harassed by other students who are prejudiced against Muggle - borns.
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.
Maybe she's tired of the often inaccurate portrayals of the characters. Fanfiction sometimes distorts the true nature of Hermione and Snape's relationship as it was in the original books. For example, some fanfictions might make their interactions overly romantic when in fact they were more complex and often adversarial in the canon.
One common scenario could be in the face of other students' bullying. Snape might subtly step in and stop the bullies without making it too obvious. For example, he could give the bullies extra assignments or detention, which would indirectly protect Hermione from their taunts.
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.
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 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.