Some good Harry Potter x Lily fanfictions focus on alternate universes. For example, an AU where Harry and Lily are both students at Hogwarts at the same time. This could lead to a whole different set of adventures and relationships. Harry could be in a different house, and their interactions would be so different from the original story. There are also fanfictions that explore the emotional side of a potential relationship between them more deeply. How Harry might feel a strange sense of attraction and confusion because of their unique connection, and how Lily might respond to this young man who is related to her in such a strange way in this new scenario.
Well, a great Harry Potter x Lily fanfiction is the one where Lily is brought back to life in some magical way. In this story, Harry has to come to terms with having his mother alive again. Their relationship is complex as she is now more of a contemporary figure to him rather than just a memory. It's fascinating to see how the author writes their interactions, as they both try to figure out how to be family in this new situation. And then there are fanfictions that rewrite the whole story with a different prophecy, one that involves Lily and Harry in a more intertwined way from the start.