Well, it could be that Lily is under some sort of spell or curse that makes her think she has to abandon Harry. In some fanfic ideas, her own past or family secrets could play a role. Maybe she discovers something about her family that makes her believe she's a danger to Harry. Another aspect could be the pressure from the wizarding community. If she's being ostracized or feared for some reason, she might think that leaving Harry is the only way to save his reputation and future in the wizarding world. It's all about exploring different creative ideas within the fanfic realm.
In some fanfics, it could be due to a threat she believes she can't protect Harry from. Maybe she thinks that by leaving him, the dark forces won't target him as much.
One reason might be that she's been manipulated. For example, if there's a powerful dark wizard who tricks her into believing that Harry is better off without her. Or perhaps she's in a situation where she has no choice but to leave to protect the greater good. Say, if staying with Harry would put the entire wizarding world at risk, she might make the difficult decision to abandon him. Also, it could be related to some sort of prophecy that she misinterprets, leading her to think that leaving Harry is the right thing to do.
Perhaps they are blackmailed. A powerful dark wizard could threaten to harm all of their friends and family if they don't abandon Harry. They may feel they have no choice but to sacrifice their relationship with their son to protect others they care about.
Perhaps in this fanfic, Harry has been influenced by dark magic or some evil entity. This external influence could distort his perception of Lily and turn his normal feelings for her into hatred. Dark magic has the power to corrupt one's mind and change relationships in unexpected ways.
One possible reason could be that she is under some sort of dark magic influence. Maybe a spell has been cast on her that makes her act out of character and abandon Harry.
I'm not sure of a specific 'lily abandons harry fanfic' off the top of my head. But you could try searching on popular fanfic websites like Archive of Our Own. They have a vast collection of Harry Potter fanfics, and you might find one with that theme there.
Maybe some fanfiction authors just like to explore extremely dark and unexpected scenarios. They want to break the traditional image of the characters to create a unique and controversial story.
In some fanfics, it could be due to a conflict over a magical object or a spell. Suppose Lily finds a powerful magical item and wants to use it in a way that Harry thinks is dangerous. This difference in opinion might lead to Harry yelling at her to stop her from doing something reckless.
Another possible theme is self - discovery. As they go through different experiences in the fanfic, they could learn more about who they are and what they want in life. For example, they might discover hidden talents or face their fears and in the process, understand themselves better.
One great fanfic could be 'Harry and Lily's Enchanted Bond'. In this story, Lily, being a fairy, might use her magical fairy abilities to help Harry in his adventures at Hogwarts. It could add a whole new dimension to the wizarding world as we know it.
Perhaps Sirius Black could also be involved. Sirius was Harry's godfather and was very much a part of Harry's life. His presence could add some interesting dynamics, like he could be the one who stumbles upon the magic that makes the meeting possible and then has to deal with the implications of Harry meeting his mother.
In a fanfic, Harry could be developed by showing how he copes with having his mother back in his life. He might have mixed feelings of joy and confusion. Lily could be developed by depicting her reaction to the modern wizarding world that Harry is a part of. She could be both amazed and a bit critical of some of the changes. This would give depth to their characters as they interact and adapt to each other.