If Hermione was James' daughter in fanfiction, Harry and Hermione's relationship would become more like a family one. They'd be closer in a different way, more like siblings.
One idea could be that the daughter inherits Hermione's intelligence and Snape's love for potions. She could be a brilliant young witch who is constantly trying to bridge the gap between her mother's Gryffindor friends and her father's Slytherin past. Maybe she uses her skills to create new potions that help solve magical mysteries in the wizarding world.
Her magic abilities could develop in a very powerful way. Since she has Harry's natural talent for magic and Hermione's studious nature, she might be able to master advanced spells at a young age. For example, she could learn Patronus charms earlier than most and have a very unique form, perhaps a combination of her parents' Patronus animals.
One possibility could be that Hermione's intelligence would blend really well with the Avengers' skills. She could use her vast knowledge of magic to assist in situations where the Avengers' technology or brute force might not be enough. For example, she could use spells to disable high - tech security systems during a mission or heal the Avengers' wounds more quickly than normal medical means.
Well, she could be highly intelligent like Hermione. Maybe she'd be a bookworm, always with her nose in a tome, eager to learn new spells and knowledge. At the same time, she might inherit Draco's sense of pride. So she'd be confident in her abilities and not easily intimidated. However, she could also have a kind heart, a blend of Hermione's compassion and Draco's loyalty to his own, which in this case would be his family.
She could be highly intelligent like both her parents. Since Hermione is known for her studiousness and Draco is quite cunning in his own way, their daughter might inherit a sharp mind. She could be very confident, perhaps a bit of a loner at times as she might feel she doesn't quite fit into the typical Hogwarts houses' stereotypes. And she might have a strong sense of justice, but also a bit of a mischievous streak from Draco.
A new dark wizard could rise. This wizard might be jealous of the fame and power that Harry has, and seeing his daughter as an extension of that, decides to target her. Maybe this new dark wizard is seeking to gain control over the wizarding world and sees her as a key obstacle in his path.
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.
Hermione could develop a more complex personality. On one hand, she has her intelligent and studious self from growing up in the wizarding world as we know it. But on the other hand, she may have a new, darker side emerging due to her connection to Bella and Rodolphus. This could lead to her being more of an outcast among her friends, and she would have to work hard to prove that she is not like her Death Eater parents.
Hermione could become more attuned to the darker aspects of magic. With Remus being a werewolf and Sirius having his own dark family history, she might explore the gray areas of magic more. For example, she could study more about the dark arts not to practice them, but to understand how to better defend against them. This would be a significant shift from her canon character who mostly stuck to the 'light' side of magic study.