Definitely Hermione. She's so intelligent and can offer a different perspective. She could help Teddy with his studies and give him advice on how to deal with the academic pressure at Hogwarts. Also, Ron. His sense of humor can lighten the mood. He could be like an older brother to Teddy, joking around with him and teaching him some fun wizarding tricks.
In some fanfictions, there might be a plot where Teddy gets into trouble at Hogwarts. Harry, as a sort of mentor figure, steps in to help him out. This could lead to a strengthening of their bond as Harry defends Teddy and teaches him the right way to handle the situation. Maybe Teddy was wrongly accused of using dark magic, and Harry uses his experience to prove Teddy's innocence.
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.
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.
In fanfiction, Harry might be protective of Teddy because Teddy is Remus and Tonks' son. Harry has a strong sense of loyalty and he may feel a great responsibility towards Teddy as he lost his parents. Also, Harry has his own experience of growing up without parents, so he doesn't want Teddy to go through any hardships alone.
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 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.
Hermione and Ron would probably be there. They are Harry's best friends, so they might be like aunt and uncle to Teddy. They could help Harry out with raising Teddy, like Hermione giving educational advice.
Most likely, Harry Potter would still be one of the main characters. Since it's in the Harry Potter universe, he's central to the story. He could be the one around whom the new bonds are formed or strengthened.