HOLD ON A SECOND.
What if Lily never died, not really, instead a Auror named Lucy took her place using poly juice potion. Lily is told to run and hide, so she does, she is told her child is safe and her husbands dead, which yes it true. She meets a nice family who let her stay with them, and after a while they decide she is part of the family. From that day forward she became Lily Granger.
;p
10 years later....
Of course Hermione only knows Lily as a Granger, not as a Potter, and thinks Lily is her mothers sister. When Hermione does get her Hogwarts letter Lily explains she is a witch too. Nobody minds she kept it a secret and Lily tutors Hermione before she leaves for Hogwarts.
But what happens....