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.
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.
A scenario could be that Lily travels back to her Hogwarts days and tries to change the way she treated Snape. This could have a huge impact on Snape's future actions and loyalties. Maybe she could have been more understanding of his difficult situation, which might have led to him not being so easily swayed by the Dark Arts.
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 great 'Lily and Scorpius Time Travel Fanfiction' is 'Time's Twist'. It has an interesting plot where they travel back to the Marauder's era and get involved in some hilarious and dangerous adventures. Their interactions with the young versions of their family members are really engaging.
One great Snape and Lily time travel fanfiction is 'Back to the Start'. It beautifully explores what would happen if they could go back in time. The author does an amazing job of capturing the essence of their relationship and the magical world. It's full of emotional moments and interesting plot twists.
One idea could be that Lily travels forward in time and meets Harry as an adult. She gets to see all the good he has done and is filled with pride. Another could be Harry traveling back in time to save Lily from Voldemort, but in doing so, he risks changing the entire future. A third plot idea is that they both get trapped in a time loop together and have to figure out how to break it while also learning more about each other.
Obviously, Lily Evans and Harry Potter are the main ones. Since it's about time travel related to them. But sometimes, James Potter can also be a major focus as he is related to Lily and Harry. Also, Severus Snape might be a key character as he has a complex relationship with Lily.
Another plotline could involve Cho gradually noticing Harry's feelings and starting to develop her own feelings in return. There might be some misunderstandings along the way that they have to overcome, like when Cho misinterprets Harry's actions or words.