One common theme is family. With Lily alive, the family dynamic would be very different for Harry. Another theme could be power and status. As a pureblood, she might have a certain influence in the wizarding world. Also, the theme of redemption could be explored. Maybe she tries to redeem some of the pureblood families' wrongdoings.
Power dynamics is a big theme. As a pureblood, Lily might have different levels of power or influence. Also, the theme of identity is common. She has to reconcile her pureblood status with her own beliefs and values, which is often explored in depth in these fanfictions. They also often deal with how her pureblood status affects her relationships, like with James Potter or Severus Snape.
Sure. There are fanfictions out there that play around with this idea. In some, Lily might use her pureblood status to influence the wizarding society in ways she couldn't before. Maybe she'd be more involved in political affairs or in reforming the pureblood ideology. Her survival could also mean that Harry grows up with a mother figure, which would have a huge impact on his character development compared to the original story.
One common theme is the strength of Lily. Despite the pain and difficulties of labor, she is often portrayed as a strong and determined woman. Another theme is the hope for the future, as the birth of her child represents a new beginning in the wizarding world.
Often, there are plotlines about him using his vampire abilities to fight against dark forces in the wizarding world. His enhanced strength and speed could give him an edge in battles.
A lot of 'Lily Luna Potter born' fanfictions have the theme of friendship. Lily has to make her own friends at Hogwarts, and these stories show how she navigates different personalities and builds lasting relationships. Magic discovery is also a theme. She might find new spells or magical artifacts that are unique to her story.
One common theme is the exploration of family ties. Since Lily is often related to other characters in the Harry Potter universe, fanfictions may focus on how her family connections influence her actions and decisions. Another theme could be the discovery of hidden powers. Lily Moon might be depicted as having some unknown magical abilities that she uncovers throughout the story.
Power and responsibility. Since Harry is a pureblood lord, he often has to deal with the power that comes with his new status and the responsibility to use it well. Another common theme is identity. He has to figure out how this new identity fits into his existing self.
There are fanfictions where James and Lily alive means that they are part of a time - travel plot. Maybe they are sent back in time to before Voldemort rose to power. They then use their knowledge of the future to prevent the dark wizard from ever becoming a threat. In this kind of fanfiction, they would interact with younger versions of their friends and family, and have to be careful not to disrupt the timeline too much. This creates a really interesting and complex story.
The theme of mentorship also comes up frequently. Sirius Black, with his wealth of knowledge and experience, is often depicted as a mentor figure to Lily Luna Potter. This can involve teaching her about magic, helping her deal with the challenges of being a Potter in the wizarding world, or guiding her through moral and ethical dilemmas.
In some fanfictions, if Lily and James were alive, they would play a huge role in the fight against Voldemort. They could train Harry and other young wizards in secret. Maybe they'd form an underground resistance group using their knowledge of Hogwarts and the wizarding world to outwit Voldemort and his followers.