They might face challenges in terms of fitting into the society of the time they travel to. Harry is well - known in his own time, but in the past, he is a stranger. Teddy, with his half - werewolf heritage, may also face discrimination in certain time periods. Also, they need to find a way to communicate with the wizards of the past. Magic has evolved over time, and some spells or magical items they are used to might not exist or work in the same way in the past.
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.
Financial challenges are likely. Teddy may not have a steady income, and things like baby clothes, diapers, and medical bills can add up quickly. She'll also have to balance her education. It's not easy to study and take care of a baby at the same time. She may need to look for alternative ways to keep up with her schoolwork, like online classes or homeschooling.
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.
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.
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.
One interesting plot could be that Harry, Ginny and Teddy travel back to the time when the Marauders were at Hogwarts. They could interact with James, Sirius, Remus and Peter, and maybe even change some small events that have a big impact later. For example, they could prevent Peter from becoming a Death Eater spy, which would change the course of the war against Voldemort.
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.
Because Teddy is an orphan. Harry, having been an orphan himself, knows the hardships. So he feels responsible for protecting Teddy.
There are also other characters like Matt Donovan who, despite being human, has a good heart and knowledge of the town. He could potentially be an ally in a different way, like providing information about the town's non - supernatural aspects. And then there's Alaric Saltzman, who has dealt with all kinds of strange things. His experience and wisdom could be valuable to Harry and Teddy.
One possible reason could be to protect Teddy. Maybe there are some threats in England that Harry wants to keep Teddy away from. For example, there could be some dark wizards or dangerous magical creatures that target Teddy for some reason.