Their relationship would likely become more complex. Since Harry has only known Lily through memories and stories, actually meeting her in person through time travel would be a shock. Lily would see Harry as a grown man, not the baby she left behind. They might develop a more equal, friend - like relationship as they face the challenges of the time - travel situation together.
I've heard of 'Time - Twisted Affections'. It delves deep into the idea of Lily having a crush on Harry in a time - travel scenario. It's full of drama and mystery as they navigate through different time periods while dealing with their growing feelings for each other.
One common theme is redemption. Harry often goes back in time with the idea of saving Lily or changing the past for the better. Another theme is the exploration of family relationships. Since Lily is Harry's mother, it gives a chance to see how their relationship could be different if they interacted in a different time and context.
One challenge could be maintaining the integrity of the original story. Since time travel is involved, any small change could have a huge impact on the future or past events. Another is the emotional aspect. Harry meeting his mother in a different context than he's used to could be very difficult for him to handle. And for Lily, seeing a son she doesn't fully know yet but has sacrificed so much for could also be quite a shock.
Well, the time travel can either bring them closer or create more distance. If they travel to a time when they're supposed to be enemies, it can be really tough on their relationship. But if they travel to a time when they can get to know each other better without outside interference, it can strengthen their bond.
One popular 'harry x lily time travel fanfiction' is 'Back in Time'. It has an interesting plot where Harry travels back in time and meets Lily in a rather unexpected way. The story does a great job of exploring their relationship within the time - travel scenario.
When they travel to the future, a powerful corporation or a group of people in control of the new world order could be their enemies. These entities might not want Teddy and Lily to expose the dark secrets of the future, like how they are suppressing the lower classes or ruining the planet. So they would try to stop Teddy and Lily from going back to their own time with this knowledge.
One interesting plot idea could be that Harry travels back in time and meets Lily before she has him. He could try to prevent her death by warning her about Voldemort. Another could be that Lily time - travels to the future and sees the world after Voldemort's defeat, getting to know the man her son has become. Or perhaps they both travel to a different time altogether, like the time of the founders of Hogwarts, and have to adapt to that era while also dealing with their own relationship and the knowledge they bring from their own times.
One common theme is the exploration of family bonds. Since Harry never really got to know his mother, time travel allows for a deep exploration of their relationship. Another theme is the alteration of the past and its consequences. Writers often play with the idea of how small changes in the past due to Harry and Lily's interaction can have a big impact on the future of the wizarding world.
One interesting plot could be Lily traveling back in time to prevent James and Sirius from playing a particularly dangerous prank that had unforeseen consequences. Another might be Harry traveling back to meet his mother as a young witch and getting to know her in a whole new way. And perhaps Lily and Harry traveling back together to change certain events that led to the rise of Voldemort, like preventing the first opening of the Chamber of Secrets.