There could also be themes related to the wizard - Muggle divide. Mr Granger, being a Muggle, might have some unique perspectives on the wizarding world that he wants Hermione to consider. This could lead to stories where Hermione has to navigate between the two worlds, trying to bridge the gap in some way. Maybe Mr Granger wants Hermione to bring some of the positive aspects of the wizarding world back to the Muggle world, or he might be worried about how the wizarding world could change his daughter in ways he doesn't understand.
Well, another option could be to ask in relevant fan communities. For example, on Reddit, there are subreddits dedicated to Harry Potter fanfiction. Post a request there asking for fanfiction about Mr Granger and his daughter Hermione. People in those communities are usually quite helpful and might be able to point you to some great stories.
One common theme is identity. Hermione has to grapple with who she really is, being the daughter of Regulus. Another is family secrets. Discovering that she is Regulus' daughter means there are likely many things she doesn't know about her family. And then there's the theme of legacy, as she has to decide how to carry on Regulus' name and any unfinished business he might have had.
She could have the power of super - intelligence as a demigod. This would make her even smarter than she already is in the original story, allowing her to understand complex magical theories and ancient spells with ease.
Another aspect to expect is a new set of relationships. Maybe she'll have some sort of connection to Greek demigods or other mythological creatures. This could lead to some exciting crossovers in the story. For instance, she could meet a Greek demigod who helps her with a particularly difficult magical task.
In such fanfictions, we might see Hermione having a different kind of connection to knowledge. Since Athena is the goddess of wisdom, Hermione could have an even deeper understanding of magic and its history. For example, she might be able to decipher ancient spells more easily or have insights into the origins of magic that others don't. This could also change her relationships with her friends. She might become more of a leader, guiding Harry and Ron with her enhanced wisdom.
Self - discovery is also a theme. Hermione often has to learn more about herself, her limits, and her potential in fanfiction. For instance, when she is faced with a new magical ability or a difficult moral choice, she has to look inward and figure out who she really is and what she stands for.
Fanfictions that explore Hermione's relationship with other characters in a different light are also interesting. For example, a story about her forming a strong friendship with a Slytherin student who she initially thought was an enemy. This would show a different side of her character and how she can overcome house prejudices.
One great fanfiction could be 'The Weasley Mentor'. In this story, Hermione forms a special bond with an older Weasley who helps her navigate the complex wizarding world post - Hogwarts. It's full of character development and interesting plot twists.
One idea could be that Hermione is raised in secrecy by someone who knew Voldemort's true identity but didn't want her to follow his dark path. She discovers her true parentage through a series of mysterious events, like finding an old diary of Voldemort's that reveals secrets. Then she has to struggle with her own identity, whether to embrace the part of her that is related to Voldemort or completely reject it and continue to fight for good.
Another theme is self - doubt. Given Hermione's usually confident nature, when she's depressed, she starts to question everything about herself. Her intelligence, her worthiness as a friend, and her place in the wizarding world. In some fics, she might even consider leaving Hogwarts because she feels she no longer belongs.