Definitely Harry. His past lives can bring so many new storylines. But don't forget about Ron Weasley. In some fanfictions, his past lives are related to great warriors or hunters in the wizarding world. Also, Dumbledore. His wisdom and power might have roots in his past lives, and exploring this in fanfiction can be really engaging. There are also some minor characters like Luna Lovegood whose past lives can add interesting twists to the story.
One interesting concept could be the idea of how past lives might affect the present characters' personalities. For example, if Harry had a past life as a great wizard in a different era, it could explain his natural aptitude for magic. Another concept might be the relationships that were different in past lives. Maybe in a past life, Hermione was a powerful sorceress who mentored Harry instead of being his peer.
Well, in some fanfictions, Harry might draw on his own experiences facing dark magic. He could use his adventures with Voldemort as real - life examples to make the lessons more vivid. For instance, when teaching about defensive spells against the Imperius Curse, he could talk about how he resisted it in his encounters.
One interesting concept could be alternate universes where characters have different fates. For example, Harry Potter might be a Slytherin instead of a Gryffindor, which would completely change his relationships and the story's dynamics.
Draco Malfoy can be quite popular too. In fanfictions about multiple lives, he could be shown as having a change of heart much earlier or being in a situation where he has to team up with Harry and Hermione from the start. His complex character and family background offer a lot of room for creative storytelling.
In the fanfiction, Harry Potter might live by himself because he is tired of the spotlight. After being in the public eye for so long, he desires a private life. He may also feel that living alone allows him to better understand himself and his powers. It could be that he has a prophecy or a personal quest that requires solitude. For example, he might be trying to master a very difficult and dangerous spell that he doesn't want anyone else to be in danger from if something goes wrong. Also, he could be exploring some dark corners of magic that the wizarding world might not approve of, and living alone gives him the freedom to do so without judgment.
I'm not sure specifically as there could be many different takes in such a fanfiction. It might involve Harry and his mom having some sort of unique adventure while living in a trunk. Maybe they discover a hidden magical world within the trunk or use it as a refuge from the dark forces in the wizarding world.
One possible reason could be the continuous loss of his loved ones. In the original series, Harry has already endured so much pain from losing Sirius, Dumbledore, etc. In the fanfiction, perhaps there are additional deaths or betrayals that push him over the edge. Another reason might be the overwhelming burden of being 'the Chosen One'. The constant expectation to save the wizarding world could finally break him.
In some fanfictions, Harry Potter might use time - turners in a more extreme way to change the past. For example, he could go back to prevent certain deaths that occurred in the original story, like Sirius Black's death. By changing this event, it could lead to a whole different chain of events in the wizarding world, such as a stronger Order of the Phoenix and a different battle strategy against Voldemort.
One consequence could be an alteration in the relationships. For example, if he saves Sirius, the relationship dynamics between him, Remus, and the other members of the Order would be different. They might be more united and effective in fighting against the Death Eaters.