Perhaps Bellatrix was forced into a situation by Voldemort. He might have wanted to create a new heir or a special pawn in his game. He could have used some form of mind control or coercion on Bellatrix to make her have a child. And this child, Harry, could have been part of a grand plan that Voldemort had. Or it could be that Bellatrix had a change of heart at some point, met someone she loved, and against all odds, had Harry, but still had to deal with her past as a Death Eater.
Maybe it was a last - minute decision by James and Lily. They might have thought that having Bellatrix as a godmother would offer some kind of protection. After all, she was a powerful witch. Or perhaps there was some sort of old family pact or agreement that made her the godmother.
One possibility is that it could be a result of some powerful magic gone wrong. Maybe she was involved in a spell that was supposed to do something else but ended up changing her hair color to white. It could also be a sign of her having a special connection to an ancient magic that is associated with white - haired beings in the wizarding world.
Hermione could be an ally. Hermione is known for her intelligence and open - mindedness. She might see the potential in Bellatrix's daughter to change and be a force for good. She could help the daughter learn about the positive aspects of magic and the wizarding world.
A possible plotline could start with Bellatrix having a strange fascination with Harry from the start. She manages to get him in her clutches and treats him like a pet. Harry, on the other hand, uses his wits and the limited magic he can access in this state to outsmart Bellatrix. There could be a lot of psychological battles between them, with Harry trying to understand Bellatrix's twisted mind while she tries to break his spirit. And all this could be set against the backdrop of the ongoing war between the good and the evil in the wizarding world.
One possible plotline could be that Bellatrix captures Harry during one of his adventures and decides to keep him as a pet for her own amusement. She might use dark magic to bind him and make him obey her commands. Another could be that in an alternate reality, Harry is somehow magically enslaved to Bellatrix from the start, and the story follows his attempts to break free from her control while she uses him for various dark purposes, like spying on the Order of the Phoenix.
One possible plotline could be that Bellatrix is somehow brainwashed or redeemed early on. Harry, seeing the potential for good in her, tries to help her further. Over time, they develop feelings for each other and end up getting married. Another could be a time - travel scenario where Harry goes back in time and changes Bellatrix's path before she becomes a full - fledged Death Eater, leading to their relationship and marriage.
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.
One possible backstory could be that he endured intense and brutal training sessions that left him scarred. Maybe he was training in a forbidden or extremely difficult jutsu that required him to push his body to the limit. Another could be that he got into numerous battles with powerful enemies without proper medical treatment afterwards, causing the scars to remain.
One scenario could be that a powerful magic spell backfires and turns Bellatrix into a small animal that Harry then takes in as a pet. Another could be that in an alternate universe created in the fanfiction, the power dynamics are so different that Bellatrix is somehow subjugated to Harry and made into his pet.
One possible plot could be that Harry discovers this long - lost family connection after some mysterious event. Maybe he finds a hidden letter or an old family heirloom that reveals the truth. This discovery could throw him into an identity crisis as he has to come to terms with being related to a Death Eater. He might then try to find out more about his family's past and his connection to Bellatrix, which could lead him into dangerous situations as he delves deeper into the dark secrets of his family.