Well, one thing that could happen is that Hermione would be exposed to a whole new family dynamic. Sirius has his own wild past and rebellious nature. He might encourage Hermione to be more daring in her magic, perhaps showing her some spells that are a bit on the edgy side but still within the bounds of good magic. Also, she could learn a great deal about the Marauders from him directly, which would give her a new perspective on the history of Hogwarts.
She could inherit some of Sirius's traits. For example, his rebellious streak might make Hermione more willing to break the rules when it's necessary to achieve a greater good.
One possibility could be that Sirius sees Hermione's intelligence and determination and decides to take her under his wing. Maybe he realizes her potential and wants to protect her from the dangers in the wizarding world that he has faced himself.
She might start to search for her biological parents. This could lead her on an adventure full of mystery and self - discovery. She could use her intelligence and magical skills to track down any clues about her origins.
In this fanfiction, Hermione might experience a whole new family dynamic. She could become closer to the Weasley siblings in a different way. For example, she might be more involved in their family traditions right from the start. Maybe she would get to participate in Weasley family meetings and have a say in family matters like a true family member.
Hermione's academic performance might improve even more. Sirius could provide her with rare books and knowledge from his family's library. For example, there could be ancient spells and magical theories that are not commonly known. This would give her an edge in her classes.
Well, Sirius would probably first ensure Hermione's safety. He'd get her away from the source of abuse. Then, he might start to heal her emotional wounds by being there for her, listening to her. For instance, if she was abused at Hogwarts by other students, Sirius could take her to Grimmauld Place and create a safe haven for her.
Often in fanfiction, Hermione might be in a situation where she feels isolated from Harry and Ron. Remus and Sirius, having been through their own share of friendship dramas, step in. Remus could use his calm and gentle demeanor to reassure Hermione that friendships have ups and downs. Sirius, on the other hand, might remind her of how strong she is on her own. They could take her on adventures around the wizarding world, showing her that there are other things to focus on besides the petty squabbles within the group, which would ultimately comfort her.
Another idea is that Sirius, as a Death Eater, has some information that Hermione needs to defeat Voldemort. She has to convince him to give it to her while also dealing with his newfound allegiance. It could be a race against time as other Death Eaters suspect Sirius of being unfaithful to their cause, and Hermione has to protect him while also getting the crucial information.
Well, Percy might gain new powers related to the Underworld. Since Hades is the god of the Underworld, he could teach Percy how to control shadows or communicate with the dead. This would make Percy an even more formidable demigod, with abilities that no one else has. He could also face a lot of challenges from other gods who might be jealous or scared of his new connection to Hades.
In this type of fanfiction, if Hermione is adopted by the Hogwarts staff, she could gain a different level of access to the school's resources. For example, if she was adopted by Flitwick, the Charms professor, she might have unlimited access to the Charms classroom to practice and experiment. This would make her even more proficient in Charms than she already is in the original story. And she could develop new spells or improve existing ones.