A father - daughter like relationship could be expected. Sirius, having lost so much in his life, might see Hermione as someone he can care for and protect, much like a daughter. He could be very protective of her and give her advice on how to navigate the complex wizarding world.
One possible plot development could be that Sirius becomes a sort of protector for Hermione. He might use his knowledge of the wizarding world and his own skills to keep her safe from various threats. For example, if there are dark wizards after Hermione due to her intelligence or her connection to Harry, Sirius could step in and fend them off.
She is likely to be her usual intelligent self. Hermione would use her vast knowledge of magic to understand the implications of being claimed by Remus. She might start researching about any historical precedents or magical laws regarding such claims.
In fanfiction, 'claimed' can have various interpretations. 'Moony' might be a character (could be a werewolf reference perhaps, like Remus Lupin's nickname in the 'Harry Potter' series). So, it might imply that in this fanfiction, Moony has a special connection or right over Hermione, like he has declared her as his in some way, be it through a magical bond or a strong emotional attachment within the fictional narrative.
One popular story might be 'Forbidden Bonds' which explores a unique relationship between them. It delves deep into their characters and how they might interact in a different way than in the original series. The author creates a lot of tension and drama, making it an engaging read.
Often, it begins with Hermione trying to help Sirius deal with his post - Azkaban trauma. She uses her knowledge and compassion to reach out to him. As he heals, he starts to see her in a new light. Their relationship then grows as they start to understand each other better. The lemon parts usually are used to explore their more passionate and private moments, which are often written in a very tender and intense way, adding a new dimension to their relationship.
In a Hermione and Sirius family fanfiction, Hermione might end up marrying into the Black family. Since Sirius was an important part of the Black family before his estrangement, Hermione could find herself involved in the family's complex politics and traditions. She could use her knowledge of magic and her Muggle - born perspective to challenge some of the old - fashioned ideas within the family. Sirius, on the other hand, could be like a mentor to Hermione, teaching her about the more secretive and dangerous aspects of the wizarding world that she might not have known otherwise.
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.
There could be a fascination with power dynamics. Sirius is an older and more experienced wizard in the Harry Potter world compared to Hermione. Fanfic writers may be drawn to the idea of how his knowledge and power could interact with Hermione's intelligence and determination, thus leading to the concept of him dominating in the fanfiction. Also, it might add an element of unexpectedness or novelty to the relationship between two such different characters.
If you're looking for a fanfiction with this specific scenario, you could try using more detailed search terms. For example, include the Hogwarts house Hermione belongs to in the search, or any specific events that might be related to it. Also, look for tags on fanfiction platforms that might lead you to the story you want. There are often tags for characters, locations, and relationships which can help narrow down the search.