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.
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.
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.
Harry in this fanfiction might be a lot more complex. His identity is no longer that of the Boy Who Lived in the traditional sense. He has to come to terms with the fact that his parents are two very different and controversial figures. He may become more of an outcast, even among his friends at Hogwarts. His sense of right and wrong could be more blurred, as he has the influence of Bellatrix's dark ways and Snape's own morally ambiguous actions. He might be constantly questioning his own nature and whether he is destined to follow a path of good or evil.
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.
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.
Another theme is the exploration of the characters' psyches. Bellatrix is a complex and often deranged character. When Harry time - travels and interacts with her, it gives the author a chance to dig deeper into what makes her tick. We see her in a different light, and Harry also has to reevaluate his own understanding of good and evil. This can lead to some very interesting character development throughout the fanfiction.
One great Bellatrix fanfiction is 'The Redemption of Bellatrix'. It shows a different side of her, exploring what could have happened if she had a change of heart. It's an interesting read as it delves deep into her character, adding layers that we don't often see in the original books.
Their relationship would likely become more complex. Since Harry has only known Lily through memories and stories, actually meeting her in person through time travel would be a shock. Lily would see Harry as a grown man, not the baby she left behind. They might develop a more equal, friend - like relationship as they face the challenges of the time - travel situation together.