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.
In a fanfiction where Hermione is Lucius' daughter, it would have a huge impact on the plot. Hermione's character arc would be completely different. She would probably have access to a lot of dark magic knowledge from an early age. Her sorting into Gryffindor would be a major shock to the Malfoy family. Lucius might try to get her to transfer to Slytherin. This could lead to a lot of internal conflict within Hermione. And the way the other characters view her would change too. Harry and Ron might be initially distrustful of her, and the teachers at Hogwarts would be on high alert as they would suspect her of having ties to the Death Eaters.
Well, if Hermione was Voldemort's daughter, it would completely turn the story upside down. For one thing, her friendship with Harry and Ron would be in jeopardy. Harry would be very suspicious of her, considering he's Voldemort's archenemy. Also, the Hogwarts faculty might treat her differently. Dumbledore, who was always vigilant against Voldemort, would keep a very close eye on her. This could lead to Hermione feeling isolated and having to really prove her loyalty to the side of good.
In fanfiction, anything can be possible. Maybe a powerful spell was cast that altered memories or blood relations. It could be part of a complex plot where Hermione was secretly switched at birth.
Well, if Hermione were Sirius's daughter in fanfiction, it would change a lot. For one, Hermione might have had a very different upbringing. She could have grown up with more knowledge about the wizarding world's secrets from Sirius. And in the Hogwarts years, her relationship with Harry and Ron might have been more complex. Maybe she would have been more involved in the Order of the Phoenix affairs earlier. Also, her magical abilities might have been influenced in different ways, as Sirius was from an old and powerful wizarding family.
One possible plot could be them dealing with the dark forces that still lurk in the wizarding world even after Voldemort's defeat. Their combined skills and knowledge as a married couple make them a powerful force against any new threats.
It can change the dynamic between characters. For example, Hermione with Lily's traits might be more nurturing towards Harry, which could change the way their friendship works. She could also influence Ron in a different way, maybe being more of a guiding figure for him.
In some fanfictions, Hermione might betray Harry due to a misunderstanding. For example, if there were false rumors spread about Harry being involved in dark magic or having a secret, evil plan. Hermione, being a stickler for the rules and the greater good in the original series, might be misled into believing these rumors and thus betray Harry.
In fanfiction, anything can be possible. Maybe a time - travel or a powerful magic spell gone wrong could be used to create this strange relationship. For example, if there was a spell that messed with people's memories and lineages, it could make Hermione seem like Bellatrix's daughter.
Well, it could change a great deal. For one, Hermione might have more access to inside knowledge about Hogwarts. McGonagall could be more protective of her, which might lead to different situations where Hermione is involved. For example, in dangerous situations, McGonagall might be more likely to step in directly to protect Hermione, rather than just as a teacher protecting a student.
For one thing, Hermione's status in the wizarding world would be very different. She would no longer be just a muggle - born witch, but a Malfoy. This would change how others view her. The Death Eaters might have a different attitude towards her, and she might be in the middle of the fight between the good side and her father's side. Also, her academic achievements might be seen in a new light, perhaps as a way to prove herself worthy to both her newfound family and her old friends.
Well, it could change a great deal. For one, Hermione might have had access to more knowledge and magic earlier. Minerva could have secretly trained her, giving her an edge in her studies. This could also change her relationship with Harry and Ron. Maybe she would be more cautious about getting into dangerous situations as she wouldn't want to disappoint her mother.