Voldemort could still be a main villain. Imagine if Teddy and Harry travel back in time and Voldemort discovers their presence. He might see them as a threat to his plans and try to eliminate them. This would lead to a lot of dangerous encounters and battles.
It could be a dark wizard from the past who somehow senses that Harry and Teddy don't belong in that time and tries to use their presence for his own evil gains. Maybe he wants to steal their future knowledge to become even more powerful.
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.
One idea could be Teddy time - traveling back to the time when Harry's parents were alive. He could try to prevent their deaths, which would create a whole new timeline. Harry would then have a very different life, perhaps growing up with his parents and without the burden of being 'the boy who lived'.
A third idea might be that both Harry and Teddy accidentally travel to a time in the past where a great magical event took place, like the founding of Hogwarts. They have to figure out how to get back to their own time while also trying not to disrupt the events that are crucial to the wizarding world's history. They might even learn some long - lost spells or magical knowledge during their time there.
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.
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.