Another motive could be related to Bellatrix's loyalty to Voldemort. Maybe she suspects Hermione of something, and this is her way of getting a reaction out of her. She could be trying to unnerve Hermione to see if she gives away any secrets or shows signs of being involved in something that Bellatrix or Voldemort would be interested in.
This moment can also lead to a more complex dynamic. Hermione might become more determined to oppose Bellatrix, not just because of their ideological differences but also because of this personal affront. And Bellatrix may see Hermione as more of a worthy opponent or a nuisance that needs to be dealt with more harshly, which would drive more interactions and conflict between them in the fanfiction.
One possible motive could be Bellatrix's loyalty to Voldemort. She might think that by kidnapping Hermione, she can get information about Harry Potter and the Order of the Phoenix. Hermione is known to be very intelligent and Bellatrix could believe she has valuable knowledge. For example, Hermione might know the location of Potter or the Order's plans against Voldemort.
One possible theme could be power dynamics. Bellatrix is a Death Eater with a lot of power and Hermione is a brave and intelligent witch. The act of Bellatrix playing with Hermione's tie can show the power imbalance between them.
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.
Well, Bellatrix is a very complex and cruel character. In the fanfiction context, she might torture Draco to assert her dominance. She is a powerful and deranged Death Eater, and by torturing Draco, she can show that she is in control. Also, it could be related to her pure - blood mania. If she thinks Draco is not living up to the pure - blood standards in some way, she may take it upon herself to 'correct' him through torture.
One possible motive could be jealousy. Bellatrix is extremely loyal to Voldemort, and Hermione is a very powerful and intelligent witch from the side opposing Voldemort. Bellatrix might see Hermione as a threat and also be jealous of the attention Hermione gets from Harry and Ron. So, she tortures Hermione to try and break her spirit and prove her own superiority.
One possible motive could be revenge. Bellatrix killed Sirius, who was very dear to Harry. So in the fanfiction, Harry might be driven by the intense desire to avenge Sirius' death.
Fanfiction often plays with established characters in new and unexpected ways. Regarding Hermione being Bellatrix's sister, it could be part of a parallel universe concept. In this alternate reality, the rules of blood relations in the wizarding world are different. The author could be using this idea to show how different circumstances can shape a character. For example, Hermione might have some of Bellatrix's fierceness if they were sisters, or Bellatrix could have some of Hermione's intelligence, which would make for a very interesting story.
One possible plot could be that Bellatrix captures Hermione and tries to torture her for information about Harry. But Hermione, using her wits and some new-found powerful magic she has been secretly studying, manages to turn the tables and kill Bellatrix in self - defense.
One reason could be pure hatred. Bellatrix despises Muggle - borns, and Hermione being a very intelligent and powerful Muggle - born would be a prime target for her wrath. Another could be jealousy. Hermione's skills and knowledge might make Bellatrix feel inferior, leading her to lash out.