One interesting plot could be Hermione's discovery of her divine heritage. Maybe she starts having strange powers or visions related to Hera. Another could be her interaction with other demigods or gods in the context of being Hera's daughter. For example, she might have a conflict with Zeus who is Hera's husband in Greek mythology.
In some fanfictions, their daughter might face a great threat to the wizarding world. Maybe a new dark wizard emerges, and she has to step up and use her combined heritage from Harry and Hermione, as well as her own skills and courage, to defeat the enemy. She might struggle with the pressure at first but eventually grow into a great hero.
One interesting plot could be Hermione Black growing up in a world where the Black family's dark reputation affects her at Hogwarts. She might struggle to prove herself different from the traditional Black family image. Another could be her discovery of a long - lost Black family magic that she has to master while dealing with the normal Hogwarts curriculum.
One possible plot could be that Voldemort's daughter is actually a good - hearted person despite her father's evil. She meets Hermione and they initially clash due to their different backgrounds. But as time goes on, they discover they both have a love for knowledge. Hermione helps her break away from the shadow of her father's name and they become friends, fighting against the remaining Death Eaters who want to use the daughter for their own evil plans.
Another plot could involve her struggle to fit in with normal society while having this extraordinary phantom - related background. She may face discrimination or misunderstanding from others, but in the end, she manages to find a group of people who accept her for who she is.
Well, in some cases, the plot might involve Hermione using her extraordinary intelligence in new and unexpected ways. For example, she could be inventing new spells far beyond what is taught at Hogwarts. Maybe she discovers a hidden aspect of magic that even the most learned wizards don't know about. This could lead to her having to face challenges from those who are threatened by her newfound knowledge.
Well, one possible interesting plot could be Hermione using her intelligence in a mercenary role to outwit her opponents. For example, she might use her knowledge of magic to create unique battle strategies.
One possible plot could be Hermione using her intelligence to find a way to repel Dementors without using a Patronus. Maybe she discovers some ancient magic or potion that can keep Dementors at bay. Another plot could involve a Dementor being somehow controlled by a dark wizard to specifically target Hermione, and she has to outwit both the Dementor and the wizard. There could also be a story where Hermione befriends a Dementor in a very strange turn of events, perhaps by understanding its true nature and finding a way to communicate with it, which would be a very unique take on the relationship between a character and a Dementor in fanfiction.
She would likely be more confident in her abilities. Since Hera is a powerful goddess, Hermione as her daughter would have a sense of divine entitlement. She might be less hesitant to take charge in situations.
Some fanfictions show Harry's daughter being a bit of a rebel. She might not want to follow the traditional Hogwarts rules, and Harry has to balance between being a strict father and understanding her need for freedom. Also, there could be a plot where she is targeted by an old enemy of Harry's, and they have to fight together to protect their family and the wizarding community.
One interesting plot could be the daughter's struggle to live up to Batman's legacy while also forging her own identity. She might face villains who target her because of her relation to Batman.