Well, it could be for the sake of creating drama. In the Harry Potter universe, the relationship between Lily and Harry is a fundamental one. By having Lily ignore Harry, it throws a huge wrench into the normal narrative. This allows the author to explore different emotional responses from Harry, his friends, and other family members. It also gives a chance to rewrite certain aspects of the story, like how Harry might become more self - reliant or turn to different people for support.
There are quite a few out there. For instance, 'Shadows of Ignorance' is a fanfic where Lily is under a spell that makes her ignore Harry. The story is interesting as it not only shows Harry's struggle but also how the other characters around them react to this strange situation. It's a complex exploration of family relationships in the context of the Harry Potter universe. Another one is 'Isolation from Mother' which specifically focuses on the psychological impact on Harry when Lily begins to ignore him. The author does a great job of portraying Harry's confusion and sadness.
One possible reason could be a plot device to create drama. Maybe the author wants to show a rift between them for the sake of a more complex storyline. For example, if there's some misunderstanding or a secret that has come to light which makes Harry act this way.
Well, I'm not sure of specific 'Severus ignores Lily' fanfictions off the top of my head. But you could also try Fanfiction.net. Sometimes the search function there might not be as precise as AO3, but it still has a vast collection. You might need to sift through a bit to find the ones that really focus on Severus ignoring Lily. It could be an interesting exploration of a different dynamic between the two characters compared to the canon.
It can add a lot of tension. For example, if Lily is used to Harry being friendly and then he suddenly starts ignoring her, it can create a sense of mystery and unease. Readers will be curious to find out why he's acting this way and will keep reading to uncover the truth.
In some darker fanfics, Sirius might be faking ignoring Harry as part of a plan to protect him from a greater threat. He thinks that by distancing himself from Harry, he can keep Harry out of danger. But of course, Harry doesn't understand this at first and is deeply hurt.
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.
Maybe to explore an alternative scenario. In the original story, Severus has a complex relationship with Lily. By having him ignore her, it can create a whole new set of character developments and plot twists. It could also be a way to show how their lives and the overall story might have been different if that one aspect of their relationship was changed.
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.
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.