If Hermione was raised by Bellatrix in a fanfiction, it would be a very dark and twisted story. Bellatrix is a cruel and insane character. Hermione could be forced to participate in dark deeds. But on the other hand, Hermione's intelligence might make her find a way to resist Bellatrix's influence secretly. She could use the knowledge of the dark magic she gains to better understand how to fight against it in the long run.
Well, in this fanfiction scenario, Hermione's personality would likely be drastically altered. Bellatrix's violent and unpredictable nature would surely have an impact. Hermione might be more aggressive and less trusting. Her magical education would also be different. Instead of the traditional Hogwarts education, she'd learn a lot of dark magic techniques. But her strong will from canon might still shine through, and she could eventually break free from Bellatrix's hold and find her true self.
If Hermione is adopted by Bellatrix in a fanfiction, it would change Hermione's life completely. Bellatrix's violent and unpredictable nature would be a huge influence. Hermione could start to question her own beliefs about magic and good vs. evil. There would be a power struggle as Hermione tries to maintain her own identity and values in the face of Bellatrix's attempts to mold her into something she's not. For example, Bellatrix might force Hermione to participate in some dark magic rituals, but Hermione would find ways to subvert them or turn them to her advantage. This could create a really unique and intense plot in the fanfiction.
If Harry Potter was raised by Bellatrix, he would likely grow up in a very dark and twisted environment. Bellatrix is a Death Eater, so Harry would be exposed to the dark arts from a young age. His values and beliefs would be very different from the ones he has in the original story. He might not have the same sense of right and wrong, and could be more inclined towards the use of dark magic.
If Hermione was raised by Voldemort in fanfiction, she might have a completely different personality. She could be more ruthless and power - hungry. Her magic might be used for more dark purposes. For example, she could be trained to be one of Voldemort's most powerful lieutenants, using her intelligence to plan his attacks and strategies.
The plot could revolve around Harry's struggle to fit in. Since he is a demigod raised by Bellatrix, he is likely to be an outcast in both the wizarding and demigod communities. The wizarding world may not understand his demigod powers and the demigod world may not accept him because of his connection to Bellatrix. He might have to go on a journey to find others like him or to find a place where he truly belongs. Along the way, he could face many challenges and enemies, both magical and perhaps related to his demigod status.
Hermione's character would become more complex. She might start off being more in tune with the dark side, but her natural goodness could still be there. It would be like a battle within her. She could have a very different style of magic, more in line with Bellatrix's chaotic and powerful style. However, she would also face a lot of internal conflict as she realizes the true nature of the things she's been taught.
Perhaps in this fanfiction, the Time - Turner malfunctions while Hermione is in the middle of a confrontation with Bellatrix. This could send them both to a different time period, maybe even one where the wizarding world is in a completely different state. They would have to find a way to work together, at least temporarily, to get back to their own time. Hermione would be reluctant to trust Bellatrix, but she might not have a choice. And Bellatrix would be both intrigued and suspicious of Hermione.
It would be very dark. Hermione would probably struggle with her identity, having a Death Eater mother. There would be a lot of internal conflict as she tries to reconcile her own values with her mother's evil ways.
Well, it could also be that Bellatrix is using the act of playing with Hermione's tie as a way to show her dominance. Since Bellatrix is a rather cruel and powerful character in the Harry Potter universe, this small act could be symbolic of her lording over Hermione, who is seen as a symbol of the resistance against the dark forces.
At first, Hermione would probably be in shock and defiance. But she'd quickly start to strategize. She'd use her vast knowledge of magic to protect herself from Bellatrix's potential outbursts.
In such a fanfiction, Harry's life would be extremely different. Bellatrix is a Death Eater, so Harry might be indoctrinated into dark magic. However, his strong will from his original upbringing might resist this. He could be in a constant internal struggle between the light he was initially inclined towards and the dark teachings of Bellatrix.