If you've read the chapters after, I definitely think Harry stole her body to keep it preserved and later bring her back. He got McGonagall to unlock his Time-Turner and I'm I guessing he used that to go back, probably when his parents were there. Remember how he looked at his watch while leading them out? That way he knew when to go back to, otherwise he never would have left the body alone like that (and this would also ensure he wouldn't meet his copy). I would guess he probably went back when he told McGonagall that he was going to say good-bye. Instead he went back, snuck Hermione's body out when no one was there, and made a copy to replace the body that would disappear over night. Remember, he was not terribly surprised or upset when he was told that the body was missing. As to what Harry did with the body, I'm not sure, though it could be that's she's inside Harry's ring now. (The ring itself, not the stone). Before I was thinking Harry's stone might have the Philosopher's Stone or the Deathly Hallow stone that Dumbledore transfigures and then gave to Harry to Transfigure, but now I'm not so sure since Dumbledore did Finite Incantation on it.