Another idea is that they travel to a time just after the end of the First Wizarding War. There is chaos and rebuilding going on. Harry and Teddy could be on a mission to retrieve a powerful magical artifact that was lost during the war. Along the way, they meet survivors of the war and learn about the true cost of the conflict. Teddy, who has grown up in a post - Voldemort world, is deeply affected by the stories of those who lived through the dark times.
One challenge could be the Time - Turner rules. If they are using a Time - Turner, they need to be very careful not to break any of the laws regarding time travel. For example, they can't be seen by their past selves or make significant changes that would disrupt the future they come from. Another challenge is the danger of the time period they visit. If they go to a time when Voldemort was active, they are at great risk of being detected by Death Eaters.
It could be an ancient dark wizard who was trapped in a time - locked state but is now released due to the time - traveling disruptions caused by Harry and Teddy. This wizard might see them as threats to his own dark designs on the wizarding world and try to eliminate them.
A great plot might be that Teddy time travels to a time when Harry was a child living with the Dursleys. Teddy could try to secretly protect Harry from some of the bullying and neglect he endured. Or perhaps Harry and Teddy travel to a time long before Hogwarts was founded to learn about the origins of magic in the wizarding world, and they bring that knowledge back to their own time to help improve modern magic.
They could time - travel to a time in Hogwarts' history when a great mystery was unsolved. Harry and Teddy could work together to figure out what really happened. For example, they might explore the mystery of the Chamber of Secrets long before Harry's own encounter with it. Teddy's inquisitive nature and Harry's experience could lead them to some amazing discoveries.
One exciting plot could be that Teddy and Lily travel back to the time of dinosaurs. They have to survive in a world full of giant reptiles and strange plants. They might discover new species that were not known before and their knowledge about dinosaurs could help modern science when they return to their own time.
Well, perhaps Teddy is in some sort of danger. Maybe he's being targeted by dark wizards because of his connection to Harry. Harry, being the heroic type, decides to fight to protect Teddy. He could use his magic skills and the support of his friends like Hermione and Ron to keep Teddy safe.
Well, perhaps in the fanfiction, Harry adopts Teddy after Teddy has some trouble at Hogwarts. Harry steps in as a father figure. He teaches Teddy advanced magic, things he learned from his own adventures. They might also have a trip around the world of magic together, visiting different magical communities and learning about diverse magic cultures.
It could be a dark wizard from the past who senses Teddy's anachronistic presence and tries to use his time - traveling knowledge for evil. Maybe a wizard who was on the verge of creating a really dark magic but was stopped in the original timeline, and now with Teddy's interference, he gets a second chance.
A possible plot in the Harry Potter Teddy time travel fanfiction could be Teddy going back to the time of the founders of Hogwarts. He could be a sort of mediator between the four founders, especially if there were some disputes or misunderstandings that led to future problems in Hogwarts. Teddy could use his knowledge from the future to try and smooth things over. This would not only change the history of Hogwarts but also have a big impact on the future wizarding world he comes from.
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.