Snape could offer Hermione a book to distract her. Knowing her love for knowledge, a rare and ancient book about magic could draw her attention away from whatever is breaking her. It would be his way of saying that there are other things to focus on and that she can find solace in learning.
Perhaps Snape would give Hermione some advice on how to deal with her emotions in a more Slytherin way. He could say 'Miss Granger, instead of letting your emotions consume you, use them as fuel. Channel your pain into determination.' This would not only comfort her but also empower her to face her problems.
Snape could also share some of his own experiences of being an outcast and misunderstood. He might tell Hermione that he too has faced hardships and that it is possible to rise above them. For example, 'I have been judged wrongly all my life, Miss Granger. Yet here I am, still standing. You are far stronger than you think.' This would give Hermione a new perspective and comfort her.
One possible reason could be the stress from her studies. Hermione is always under a great deal of pressure to excel academically. Maybe she reached a breaking point due to the overwhelming workload. Another reason might be some kind of conflict with her friends. For example, if she had a big fight with Harry or Ron over something serious, it could leave her feeling broken.
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 some fanfiction, it could be a plot device to build a different kind of relationship between them. Maybe the writer wants to show a gradual change in their relationship from the usual teacher - student one in the books. The inuendos could be the start of that transformation.
In many fanfictions, Snape saving Hermione from Voldemort can be a very interesting plot. It could be due to Snape's complex character. He was a double - agent in the original series. So, in fanfiction, his loyalty to Dumbledore might drive him to protect Hermione, who is an important part of the good side against Voldemort.
One possible reason for Snape making inuendos to Hermione in fanfiction could be to add an element of unexpectedness. In the original series, their relationship was mostly one of teacher - student with a bit of animosity. Fanfiction writers might use inuendos to explore a different side of their relationship, perhaps a more complex or even a slightly flirtatious side. This can make for an interesting read as it goes against the typical perception of their interactions.
He may show a rare moment of kindness. Given his own past traumas and experiences, he might empathize with Hermione on some level. He could offer her some words of wisdom or even some form of magical assistance to help her get better.
One possible reason could be that she was attacked by some dark creatures while on a solo adventure in the Forbidden Forest. The forest is full of dangers, and Hermione, being brave and curious, might have ventured too deep and got injured by a creature like a werewolf or a giant spider.
There probably is. The world of fanfiction is vast and diverse. Writers like to play around with different character relationships. For 'Hermione is Snape's mate' fanfiction, it could be interesting to see how they handle the age difference, their different personalities, and the fact that they are from different Hogwarts houses. It might not be a common pairing in the official Harry Potter lore, but in fanfiction, anything is possible.
In some fanfictions, Snape is made out to be Hermione's father. Fanfiction writers often like to play with established relationships and create alternate backstories. This idea might be appealing as it adds a new layer of complexity to both characters' stories. It can lead to different interpretations of Snape's actions towards Hermione in the original series, like perhaps his strictness was a way of protecting his secret daughter in a way.