Well, if Hermione was Lucius' daughter, it would completely turn the story upside down. First, Hermione's loyalty to the Order of the Phoenix would be in question. She might have been raised with pure - blood supremacist ideals which would make her relationship with Harry and Ron very different. Instead of being friends, they could be enemies at first. Also, her academic achievements at Hogwarts might have been used for more sinister purposes by Lucius. He could have used her intelligence to further the Death Eaters' goals, like finding ways to infiltrate Hogwarts more easily or uncovering secrets of the Order.
One common theme is redemption. Lucius is often portrayed as a character in need of redemption, and Hermione is the one who can help him see the error of his ways. This can involve him gradually changing his views on Muggle - borns and the Dark Arts.
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.
If Hermione was Lucius' daughter, her relationship with other Slytherins would be complex. Some Slytherins would likely welcome her as a pure - blood heir of the Malfoy family. However, she might also face some jealousy. Her intelligence could be seen as a threat to some of the more power - hungry Slytherins. And given that Hermione has a strong moral compass, she might not fit in completely with the more devious Slytherin crowd. She could be an outcast among them while still being part of the house.
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.
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.
Hermione's character would likely experience a great deal of conflict. On one hand, she has her own strong values and beliefs which are in direct opposition to the Malfoy family's pure - blood ideology. But on the other hand, she might be forced to adapt to her new living situation. She could become more guarded and learn to play the political games within the wizarding society as she tries to survive in the Malfoy household.
In this type of fanfiction, it could start with Hermione being an orphan for some reason. Lucius, seeing her magical abilities and intelligence, decides to take her in. At first, Hermione would be very much out of place in the Malfoy Manor. However, as time passes, she might start to change the Malfoy family's views little by little. Lucius would have to deal with the reactions of other pure - blood families, and Hermione would face challenges from both the Malfoy family's expectations and the outside world's perception of her being in the Malfoy household.
One idea could be that they are forced to work together on a magical research project. Hermione's vast knowledge and Lucius's experience in the wizarding world create an interesting dynamic. They start off hating each other but gradually come to respect one another's abilities.
One common theme is redemption. Since Lucius has been a Death Eater, many fanfics explore his journey towards redemption with Hermione often being a catalyst for this change. For example, her values and her belief in second chances can inspire Lucius to turn his life around.