One possible fanfiction could involve Bellatrix going back in time to try and change the outcome of the Battle of Hogwarts in her favor. Maybe she uses time - turners or some dark magic to travel back. She could attempt to kill Harry Potter as a baby before he even becomes a threat.
In a time - travel Bellatrix fanfic, she might go back to the time when she was first introduced to the Dark Arts. This could show a different side of her, perhaps a more innocent - looking but still dark - inclined Bellatrix. She could interact with younger versions of other Death Eaters and form different alliances that change the course of the future in the wizarding world.
She could travel back to the time when Voldemort was still in hiding as a young Tom Riddle. Bellatrix might try to influence him in a different way, perhaps making him even more ruthless. This would change the course of the wizarding world in a very different direction.
In a time travel scenario, their relationship could become a complex love - hate relationship. Bellatrix may find herself attracted to Harry's courage and determination in a different way than she's used to. Harry, while disgusted by Bellatrix's actions in the original timeline, may start to see a different side of her when they are in a different time period. This would be a very strange and morally ambiguous relationship, full of internal and external conflicts.
Sure, there is. A lot of fanfiction writers are drawn to the idea of pairing Harry Potter and Bellatrix because of the extreme contrast between them. One story I read had Bellatrix under a spell that made her forget her allegiance to Voldemort, and she ended up helping Harry. This led to a series of events where they started to see each other in a different light, and the fanfiction explored their evolving relationship in great detail.
Bellatrix Black, in a fit of desperation to change the outcome of the Battle of Hogwarts for the Death Eaters, stumbled upon a time - travel spell. She traveled back to the time when Harry Potter first arrived at Hogwarts. She planned to kill him right away, but she was thwarted by Dumbledore, who seemed to sense something amiss. Dumbledore, with his great power, managed to trap Bellatrix in a time - loop for a while until she finally managed to break free. But by then, she had missed her chance to harm Harry and had to return to her own time, realizing that time itself was a powerful force that couldn't be easily manipulated.
Well, a frequent theme is the exploration of good vs. evil in a more nuanced way. Since Harry represents good and Bellatrix is a symbol of evil in the original series. Fanfictions might also focus on the idea of forbidden love, as their relationship goes against the natural order of things in the Potterverse. There could be themes of manipulation too, as Bellatrix is a master of that and how Harry could deal with it if they are in some sort of relationship.
She could go back in time to meet Tom Riddle when he was still a student at Hogwarts. This would be an interesting encounter as she could try to influence him in different ways compared to what actually happened in the original story.
Hermione could be included. She might be suspicious of Harry's interactions with Bellatrix and try to interfere or help, depending on the situation. Her intelligence could also be crucial in understanding any strange magic related to Bellatrix.
One common theme is redemption. Sometimes Bellatrix is given a chance at redemption through the time - travel aspect, like she might see things differently and change her ways. Another theme is the power of knowledge. Harry, with his knowledge from the future, has to figure out how to use it in the past to deal with Bellatrix and the challenges of that time.