Obviously, Harry himself. His memories of his parents, his time at Hogwarts, and his battles against Voldemort could be explored in more detail. Also, Dumbledore. His past is full of mysteries, and his memories could hold secrets about the history of the wizarding world and his own actions against dark magic.
The Marauders (James, Sirius, Remus, and Peter). Their memories of their time at Hogwarts, their pranks, and their friendship could be a great focus. We could see their memories of becoming Animagi, for example. Also, Ginny Weasley. Her memories of being possessed by Voldemort's diary and her growth as a witch would be interesting to explore.
Luna Lovegood. Her unique perspective and experiences could lead to some interesting memories. Maybe her memories of her mother, or her encounters with magical creatures that others don't believe in. And Severus Snape. His memories of his love for Lily, his time as a Death Eater and his double - agent role are rich with potential for a fanfic.
Winky the house - elf. She has a lot of potential. In a fanfic, we could explore her life after being freed from the Crouch family. Maybe she has some unique knowledge about the Dark Arts or the Triwizard Tournament that she's been hiding. Her loyalty and her struggle with her new - found freedom could be really interesting to write about.
I think Neville Longbottom is a bit underrated in this book. A fanfic could focus on his growth during this time. He could have some secret knowledge about the plants in the Hogwarts grounds that helped in some way during the events. Also, Seamus Finnigan could be interesting. What if he had some family connection to the events in the Prisoner of Azkaban? Maybe his family knew something about Sirius Black or the Dementors. And Crookshanks is another character. A fanfic could be about his adventures around Hogwarts, and how he was actually more intelligent than people thought and was trying to solve the mystery of Sirius Black himself.
One underrated character for Harry Potter OotP fanfiction is Seamus Finnigan. He had his own doubts about Harry in the original story. You could expand on that. What if he had a change of heart in a more in - depth way? Maybe he discovers something that makes him fully trust Harry again. Also, Parvati Patil could be a focus. She could have her own side story about her Divination skills and how they might play a role in the bigger battle against the Dark Lord.
Definitely Firenze. He made an appearance in this book, but in fanfiction, he could be more involved. Since he has unique insights as a centaur, he could play a key role in helping Harry understand the prophecies or the true nature of the events related to Sirius. Also, consider Madam Pomfrey. She's often overlooked, but she could be a great source of knowledge about past students and events at Hogwarts that might be relevant to the situation with Sirius.
Neville Longbottom is often underrated. In a fanfiction, he could have a more central role within the castle. Maybe he discovers a hidden herb garden in the castle grounds that has magical plants with the power to heal any wound. Another could be Luna Lovegood. She could find a secret room in the castle filled with strange and wonderful magical creatures that only she can communicate with.
The children of the original main characters are often focused on. For example, James, Albus, and Lily Potter. Their experiences at Hogwarts and how they live up to their parents' legacies are common themes.
Polly Chapman is quite underrated. In fanfiction, you could explore her life more. Maybe she has a secret talent for magic that no one knows about. Another could be Craig Bowker. He could have his own story, perhaps he's involved in some kind of underground magic club at Hogwarts that's not related to the main plot of the cursed child.
Colin Creevey could be a great character to focus on. He's often seen as just a bit of a fangirl - type character, but in fanfiction, he could have a hidden talent or connection to the Chamber. Maybe he has a family heirloom that gives him some special insight into the secrets of Hogwarts. Another is Myrtle Warren. There's so much more to her story than just being the ghost in the bathroom. She could play a more important role in the events of the Chamber, like having some knowledge that she didn't share in the original story.
Harry Potter is usually still a main focus. After all, he's the hero of the original series. Hermione Granger and Ron Weasley are also often central characters. Their friendship with Harry and their own relationship are likely to be explored further.
Harry Potter himself is often a central focus. Since he's the main character in the original series, it makes sense that in time loop fanfics, his experiences in the loop are explored. Hermione Granger is also common. Her intelligence and problem - solving skills are often put to the test in the time loop scenarios. And of course, Ron Weasley, as Harry's best friend, is frequently involved as well.