In the fanfic, Bellatrix might use her magic to protect Harry from his abusers. Maybe she realizes the wrongness of the abuse and decides to be his unexpected savior. She could use her dark magic skills to intimidate those who harm Harry, creating a sort of dark - but effective - protection around him.
The Dursleys are the most likely abusers as in the original Harry Potter story, they were the ones who mistreated Harry. They could be the ones from whom Bellatrix saves Harry in the fanfic.
It could be an ancient dark magic artifact. The artifact has a hold on Bellatrix, and in a way, it's the 'enemy' that Harry has to defeat to save her. The artifact might be trying to use Bellatrix to carry out some dark plan, like opening a portal to a world full of dark creatures. Harry has to figure out how to break the connection between Bellatrix and the artifact while also fending off the artifact's attacks.
In a 'Harry Potter Saves Bellatrix' fanfiction, perhaps Bellatrix is in a situation where she's about to be killed by other Death Eaters who think she's a liability. Harry, having a sense of justice and seeing the potential for redemption in her, steps in to save her. He could then try to show her the error of her ways, using his own experiences of losing loved ones to Voldemort's ideology. Bellatrix might initially resist, but gradually come to understand Harry's point of view.
For Bellatrix, she could start to question her loyalty to Voldemort. Her interactions with Harry might make her realize that there's more to the world than just blindly following the Dark Lord. As for Harry, he could become more aware of the true nature of the Death Eaters at a very young age, which would shape his future battles. He might also develop a sense of pity for Bellatrix in a way, understanding that she's a victim of Voldemort's ideology too.
There should not be any such fanfic. The Harry Potter series is about friendship, courage and magic, not about abuse. This kind of concept goes against the positive values that the original work promotes. It is also unethical and can be very disturbing to the fans who love the original story.
Well, perhaps there's a magical accident that causes Bellatrix and little Harry to be stuck in a time - loop or a magically sealed room together. At first, Bellatrix is enraged and wants to kill Harry immediately. But as time passes, she starts to notice Harry's courage and tenacity, which reminds her of the qualities she once admired in herself or others. Harry, in turn, tries to find a way to escape while also trying to figure out how to use Bellatrix's internal conflict to his advantage.
In a Bellatrix and child Harry Potter fanfic, the author might explore the idea of Bellatrix having a change of heart towards Harry. It could start with her seeing some of James Potter in the young Harry. This could lead to a series of events where she tries to protect Harry from other Death Eaters, which would be a huge deviation from her character in the original series. However, it would make for a very interesting and unique fanfic.
One great Bellatrix fanfic is 'The Redemption of Bellatrix'. It shows a different side of her, exploring the possibility of her having a change of heart. It's really interesting to see how the author weaves a new story around her character.
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.