This prompt is slightly inspired by the movie Ghost
Harry dies during the face-off with Voldemort in the forest but comes back as some sort of spiritual being (which I'm likening to what Voldemort became after the Killing Curse backfired and he spent years in the Albanian forest as a spirit) but Harry thinks that he is a ghost. He hasn't come back (or remained behind, however you want to see it) because he fears death - it’s quite the opposite for him in that aspect - but he is still there because the living isn’t done with him yet or he isn't done with the living. Either way.
He begins to follow his friends around, Ron and Hermione (and Ginny is there too I guess), but they can't see him. Ron and Hermione's relationship never gets started, they both grieve Harry's death in different ways, he goes straight into the Ministry with vengeance to track down every last Death Eater on the planet, Hermione goes to Australia to reinstate her parent’s memories and perhaps stays there for the summer when they tell her they aren't returning to England. Ginny, however, goes down a much darker path, and the rate of her going through boyfriends or lovers or whatever increases so drastically, that even Harry - who is not restricted in his ability to travel - loses track of her.
He returns to Hogwarts after that to seek the advice of the other ghosts of the school but becomes frustrated when he realises that they can't see him either. That's when he realises that he is not just a ghost, but a spirit trapped in a separate space completely. He doesn't know what else he can do, so when the new school year arrives, Harry is drawn to Hermione, the person he always turns to for help. But, of course, she can't see or hear him, and he can't do anything to get his attention, so he just watches.
He feels like he is being creepy when he follows her everywhere, classes, mealtimes, the library, her dormitory, the bathroom... but he also realises just how clearly depressed she still is about his passing, her head is always down, her shoulders always slouched, her schoolwork well below exemplary, and she seems be crying all the time.
One day, in a shared class with Ravenclaws, Luna Lovegood says hello to Hermione in passing, and also says hello to Harry. Hermione reminds Luna that Harry is dead and she's like "oh yeah, silly me, sorry," and walks away. Hermione shakes her head and thinks nothing of it, but Harry couldn't believe that the blonde-haired girl had just made eye contact with him, when not a single other being had.
It takes a lot of convincing before Hermione accepts that what Luna is seeing really is Harry. She asks a question only Harry would know the answer too or something like that. The three of them have to work together – which is difficult when Luna stares off into space a lot, forgets that Harry and Harmony can’t hear each other, and makes some very absurd suggestions – to find out why Harry is stuck between life and death and how to fix it.
The rest of what happens is opened to interpretation, but my belief is that he is either reunited with his body and becomes a physical being that’s able to live again, or they give him a proper burial, and he’s able to “move on”.