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.
Well, it would change a great deal. Bellatrix being Harry's godmother would mean a completely different relationship with the Dark side. Instead of being a feared Death Eater out to kill Harry, she might be in a position of protecting him, which would create a very complex and strange dynamic within the wizarding world. It could also lead to some really interesting plot twists where Bellatrix has to choose between her loyalty to Voldemort and her role as Harry's godmother.
A potential godmother could be a long - lost relative of Lily Potter. Maybe a sister who was thought to be dead but actually survived and has been living in seclusion. She would be very powerful and have a strong connection to Harry through his mother.
Well, it could also be that Bellatrix is simply bored with the reality of her world in the Death Eater realm. Fanfiction provides an escape into a different world where she can interact with Harry in ways that she may not be able to in the official story. It gives her a chance to be part of new and exciting (to her) adventures that revolve around Harry, even if they are just fictional interpretations created by fans.
She might influence him by giving him moral support. In the face of the dark forces, having his godmother's support could boost Harry's confidence.
One reason could be revenge. Bellatrix might want to get back at Harry for causing problems for Voldemort and the Death Eaters. Kissing him could be a form of humiliation. Another reason could be a power play. By kissing Harry, she shows that she can get close to him and has some sort of control over the situation.
Maybe she wants to humiliate him. By treating him like a baby, she can make him seem weak and powerless.
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.
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.
There might be. Fanfiction is a place where creativity knows no bounds. Writers might have thought that having Amelia Bones as Harry's godmother could add an interesting dynamic to the story. Maybe she could protect Harry in ways that others couldn't, or perhaps there could be a whole new backstory involving her relationship with James and Lily. It could also lead to different plot developments in the face of Voldemort's threat.