There are indeed fanfics like that. It's a very creative idea. Imagine the implications for Hermione's character. She would inherit different traits from both Snape and Minerva. Snape's intelligence and his complex nature, along with Minerva's wisdom and strictness. This would change her backstory and how she interacts with other characters in the fanfic world.
Well, first of all, her personality might be a bit different. Maybe she'd be more reserved like Snape in some ways, but also have Minerva's confidence. In terms of the plot, she could have a different role in the fight against Voldemort. Since her parents are Snape and Minerva, she might be more involved in the inner workings of Hogwarts' defenses. And her relationship with the other students, especially Harry and Ron, would likely change. She might be more of a leader from the start, given her powerful parentage.
In most fanfics, the status of a character's parents can be quite variable. For this 'Harty' in the Hermione fanfic, without having read it, it's difficult to say. It could be that the author decides to keep them alive to have a support system for Harty or perhaps to create a contrast with Hermione's own family situation. If the fanfic is about Harty's adventures in the wizarding world similar to Hermione, having living parents could mean they are either muggles who need to be protected from the wizarding threats or they could be wizards themselves and be part of Harty's journey.
There are probably fanfictions out there with this premise. Fanfic authors love to play with the established canon of a story. By making Minerva McGonagall Hermione's mother, they can rewrite many aspects of Hermione's life. Maybe she would have grown up with a different set of values, or her magical abilities could be seen in a new light. It also has the potential to create new conflicts and reconciliations within the Hogwarts community.
In fanfiction, Hermione and Minerva could have a wonderful friendship. Hermione's thirst for knowledge would mesh well with Minerva's vast wisdom. They might often be found in the library together, discussing ancient spells or the latest magical theories. Their friendship could also be a source of support during difficult times at Hogwarts, like when the school is under threat.
One possible scenario in such a fanfic could be that during detention, Snape makes Hermione clean the potions classroom. At first, Hermione is indignant but as she cleans, she discovers some old potions notes that Snape had written as a student. This leads to an unexpected conversation between them about the complexity of potions. Snape is initially cold but gradually warms up as he realizes Hermione's genuine interest in the subject.
Maybe in the fanfic, Snape gives Hermione detention for accidentally disrupting his class. But instead of a typical punishment, he assigns her to help him organize his personal library of dark arts books. As they work together, Hermione begins to understand Snape's complex past and Snape starts to see Hermione in a new light, beyond just being a know - it - all Gryffindor.
One great Snape Hermione fanfic is 'The Tapestry of Time'. It weaves a complex story where Snape and Hermione are forced to work together on a time - sensitive magical project. Their relationship develops gradually from mutual suspicion to a grudging respect and eventually something more. The characterizations are really well - done, staying true to their canon personalities.
A theme of friendship is also great. Hermione could be the one to break through Snape's cold exterior and they become unlikely friends. They could exchange books, discuss magical theories, and help each other grow. Also, a theme of them being part of a team in a magical competition can be interesting. They would have to rely on each other's strengths to succeed.
Snape might initially be taken aback. He's used to students being intimidated by him, so Hermione's outburst could shock him into momentary silence. Then he might retaliate with his usual sarcasm, trying to put her in her place.