Obviously, Harry Potter himself is a central character. His wife, whether it's Ginny or in some alternate scenarios someone else, would also be very important. Then there are his friends like Ron and Hermione. They would likely be the baby's godparents or play important roles in the baby's life. Dumbledore, even though he's passed away, might still have an influence through his wisdom left behind. And of course, there could be new characters created specifically for the fanfic, such as a nanny or a tutor for the baby with some unique magical skills.
Harry's family members would be in the picture. His in - laws, like Ginny's family. Neville could also be a part of the story, maybe as a teacher at Hogwarts when the baby grows up. The Malfoys could be an interesting addition too. They might have a change of heart and be curious about Harry's baby. The baby's own peers as it grows up, other children in the wizarding world, would be potential characters. There could be rivalries or friendships formed right from a young age. And there are always the magical creatures, like Hedwig's descendants, which could have a special connection to the baby.