Another possibility is that his time - traveling could create some sort of time paradox. This might lead to all sorts of strange and unexpected consequences. His presence in the past could disrupt the normal flow of magic and time, and he'd have to find a way to fix it before everything goes wrong.
His character would likely become more mature. Seeing the past and knowing the future would make him more responsible. He might not be as reckless as he was before.
One interesting plot could be Harry going back in time and preventing certain tragedies from happening in the wizarding world, like stopping Voldemort's rise before it really began. Another could be him changing the relationships between characters, for example, making Snape and James Potter become friends early on.
Harry himself is obviously a very popular character in these fanfictions as he is the one being sent back in time. But also, characters like Sirius Black are often popular. Fans like to explore how Sirius would react to Harry being in the past, and how their relationship could develop differently.
Definitely Harry himself. Since he is the one being sent back in time, he is central to the story. His actions and decisions in the past are what drive the plot forward.
One interesting plot twist could be that Harry goes back in time and accidentally changes the sorting of some of the characters. For example, Draco Malfoy could end up in Gryffindor instead of Slytherin, which would completely change their relationship and how events unfold in the wizarding world.
A great fanfiction idea could be that James Potter time - travels during a crucial battle in the wizarding world. He travels back to a time when the dark forces were just starting to rise. With his future knowledge, he could start preparing the wizarding community earlier. He could train young wizards and witches in advanced defensive magic, and try to change the course of history so that the big battle he left behind would have a different, more positive outcome.
Revan sent back in time might completely change the course of galactic history. He could influence key characters in the past. Maybe he'd train a young Jedi who would become a great hero earlier than in the original timeline. Or he could interfere with the Sith teachings, leading to a different breed of Sith or even weakening the Sith order as a whole.
A plotline could be that James Potter comes back to life but with some sort of special power or knowledge. Maybe he has a connection to an ancient magic that was lost. This leads him on a quest to discover more about it. He might also have to deal with the emotional turmoil of seeing his friends and family having moved on without him. His relationship with Lily, if she's also brought back in some way, or how he copes with her being gone can also be a major part of the fanfiction.
One common theme is self - discovery. Since Harry is sent back in time, he gets a chance to see his past self and the world in a new light, which often leads to him learning more about his true nature. For example, in some fanfics, he realizes his hidden strengths or weaknesses. Another theme is the alteration of the timeline. His presence in the past can change small or large aspects of the future he came from, and the fanfics usually explore the consequences of these changes.