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.
In certain fanfictions, Harry is shown as being protective of Teddy during magical training. He would be extra careful and supervise Teddy closely to prevent any accidental magic from going wrong and harming Teddy. Also, when it comes to Teddy's emotional well - being, Harry might be the first one to notice if Teddy is sad or troubled. For instance, if Teddy is bullied at school, Harry would take immediate action to stop it and comfort Teddy.
In some versions, Neville could also be one of Harry's protective mates. Neville, who grows in courage throughout the series, would be willing to stand up against any threat to Harry. And Luna, with her unique perspective and magical abilities, might also be part of Harry's protective group.
Maybe Blaise sees something special in Harry. In fanfiction, it could be that Blaise admires Harry's courage or his unique abilities. He might feel a sense of kinship with Harry despite their differences.
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.
Well, it could be that in the fanfiction's plot, Harry has some special qualities or is in a dangerous situation. Edward, being a character with his own set of values, decides to be protective. For example, if Harry has some sort of power that others might want to misuse, Edward would step in to safeguard him. It might also be related to their shared experiences or a bond that has developed between them over the course of the story.