I'm not sure specifically about 'harry possesses lily' fanfics being extremely popular. There are so many fanfics out there that it's hard to single out ones with this exact concept as being widely known. However, some niche fanfic communities might have some interesting takes on it.
It could mean a fan - created fictional story where Harry has some form of possession related to Lily. Maybe it's a magical possession in the context of the Harry Potter universe, like he has an object that belonged to Lily or some special connection to her.
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.
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.
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.
Well, a fanfic about Harry and Lily Brerd could explore their relationship in various ways. Maybe it delves into their family bonds, or perhaps it creates some unique adventures for them. It could also show how they interact with other characters in the fan - created world.
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.
In the fanfic, Lily's fairy heritage could make her a target for dark forces. Harry, being the hero he is, would then vow to protect her. Their relationship would grow as they face numerous challenges together, and Harry might learn some new spells or techniques from Lily's fairy magic along the way. This would not only strengthen their bond but also make Harry a more powerful wizard.
Since this is a fan - created concept that goes against the fundamental relationships in the 'Harry Potter' world, it's hard to define what a specific fanfic like this would be about. Fanfics should generally stay within the boundaries of the original world's ethics and relationships.