One idea could be exploring the lives of the characters after the events of the original trilogy. For example, what if Katniss had to deal with post - traumatic stress in a new way? Another could be creating a new Hunger Games scenario in a different part of Panem, with new tributes and different rules. Maybe a fanfiction where the Capitol is rebuilt in a more just way and the former districts have to adjust to this new political situation.
You could write a fanfiction about the children of the main characters. Imagine the challenges they face, growing up in a world that's still recovering from the horrors of the Hunger Games. Or perhaps a story where a new threat emerges that's similar to the Capitol's tyranny, and the heroes of the past have to train a new generation to fight. There could also be a fanfiction focused on the relationships between the different districts, how they're rebuilding trade and cultural exchanges.
A cool fanfiction concept could be to tell the story from the perspective of a minor character in the fourth book. Maybe a Capitol citizen who starts to question their way of life after the fall of the old regime. Or a story that dives deeper into the mythology of Panem, uncovering secrets that were never revealed in the original books. It could also be interesting to write a time - travel fanfiction where a character goes back in time to prevent some of the tragedies that occurred during the Hunger Games.