One possible plot could be that Harry starts to show signs of Lily's temper in his interactions at Hogwarts. For example, when facing bullies like Draco, instead of just ignoring or reacting calmly as he sometimes does, he might flare up more quickly due to his inherited temper. This could lead to more confrontations and also some misunderstandings among his friends who are used to his more laid - back side.
His relationships would be more turbulent. With Ron, they might have more arguments because Harry's temper could cause him to overreact to small things. In his romantic relationships, if he has any, his temper could be a turn - off or a source of conflict. For example, if he's dating Ginny, his outbursts could lead to fights that they'd have to work hard to resolve.
One possible plotline could be Harry getting into more arguments with authority figures at Hogwarts. Just like Lily might have when she was a student. For example, he could have more clashes with Snape, which would add a new layer of tension to their already strained relationship.
It could be Draco Malfoy. Malfoy is always taunting Harry and his friends. His constant insults and attempts to undermine Harry might finally push Harry over the edge and make him lose his temper.
Well, without reading the specific fanfiction, it could be that his mother's temper was kind - quick to protect and stand up for those she loved. Maybe she was fiery when it came to facing injustice, and Harry inheriting that means he might also be passionate about defending his friends and what he believes in.
Well, it could make his relationships more volatile. His friends might be taken aback by sudden outbursts similar to Lily's. They may need time to adjust and understand that it's not really Harry being mean, but rather this new aspect of his personality. For example, Ron might initially be confused and a bit hurt, but as he realizes it's part of this new 'Lily - like' Harry, he could learn to handle it better and be more patient.
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.
Well, without reading the specific fanfic, it's hard to say exactly. But generally in Harry Potter fanfics, Lily could have all sorts of plans. It might be related to protecting Harry, or perhaps something to do with the wizarding world's future. Maybe she has a plan to deal with Voldemort in a different way than in the original story.
Well, in some fanfics, Harry's fame might make people around him act differently. They could be overly cautious or overly eager to please. His temper, if it's written as easily flared in fanfic, could scare people because they don't want to be on the receiving end of his anger. For example, if he gets angry at someone for a small mistake, it might make others fear getting close to him.
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.