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.
Another plot might be that they travel to the future. They could see how Hogwarts has changed, the new magical inventions and the different social structure in the wizarding world. They might also meet their descendants and learn about their own legacies. It would be a fascinating exploration of the unknown future of the wizarding world.
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.
Perhaps Teddy gets into trouble at Hogwarts, and Harry and Ginny have to step in to sort things out. It could involve Teddy being wrongly accused of something, and they use their skills and influence to clear his name. Or Teddy discovers a secret room in Hogwarts that only he can access, and he enlists Harry and Ginny to explore it with him.
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 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 plot might involve Teddy getting into some sort of trouble in New York, like accidentally using magic in front of Muggles, and Harry has to come to his rescue and figure out a way to cover it up without breaking the Statute of Secrecy. It could lead to a lot of adventures as they try to avoid the attention of the Magical Congress of the United States of America.
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.
Luna Lovegood would be a great addition. Her unique perspective on magic could offer new insights to Teddy. Also, Ron could be included as Harry's best friend. His humorous and sometimes clumsy nature could add some light - hearted moments to the story.
One interesting plot could be that Harry and Amelia travel back in time to the founding of Hogwarts. They get to meet the four founders and witness the creation of the school's magical protections. Another plot might be them traveling to a time when a great dark wizard was at the height of his power and they have to find a way to stop him without changing too much of the future.
One interesting plot could be that Harry and Hermione travel back in time to change a major event at Hogwarts, like preventing the opening of the Chamber of Secrets. They might use their knowledge from the future to outwit Tom Riddle before he becomes Voldemort.