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 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.
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.
When Harry and Teddy time travel, Harry could become more open - minded. He has been through a lot in his own time, but seeing different cultures and ways of life in other time periods would shatter some of his preconceived notions. Teddy would mature faster. Imagine traveling to a time of great turmoil in the wizarding world. Teddy would have to step up, make decisions, and face consequences. This would transform him from a somewhat carefree young wizard into a more responsible and strategic one. Harry, too, would learn from Teddy's growth and vice - versa, strengthening their bond and further developing their characters.
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.
In some Harry Potter fanfictions about Teddy Lupin's time travel, he might go back to prevent certain tragedies. For example, he could try to save his father Remus Lupin from the final battle. He might use his knowledge of the future to change the course of events, perhaps by warning the Order of Phoenix earlier about Voldemort's plans.
One interesting story could be where Teddy Lupin travels back in time to meet his parents. He gets to experience their lives firsthand and maybe even change some small things that have a big impact on his own future. It could be a heartwarming tale of family connection.
Well, their relationship would be complex yet beautiful. Since Teddy is just a baby, Hermione would have to take care of all his basic needs. As they travel, Hermione would start to teach Teddy about different magical things she discovers in each time period. Teddy would be curious and his innocent questions would make Hermione think more deeply about magic. This exchange would deepen their relationship. Also, as they face challenges together, like when they are misunderstood in a certain time period or when they have to face powerful wizards, they would become closer. Teddy would see Hermione as his hero, and Hermione would see Teddy as a precious charge she needs to protect and nurture.
Teddy, being a young character, would grow up very quickly in a time - travel story. He'd learn about the great sacrifices made by his family and the importance of loyalty. He could also develop new magical abilities by being exposed to different magical teachings from different eras. His character would transform from a somewhat naive young boy to a more mature and capable wizard, with a much broader view of the wizarding world.
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.