This is the song that this refrain--the music box snippet--reminded me of: The Zombies 'The Way I feel Inside' (also Taron Egerton's version in Sing).
It's not identical, I know; but, there's a few notes that match this song and how--in both versions--the voice follows the melody.
It's this section that is reproduced in Harry potter and the Half-Blood ...
Apparently it isn't from a popular song. It was composed by Nicholas Hooper for the scene Ron In Love and is titled Moonlit Night The full song can be heard here:
Harry Potter and the Half-Blood Prince (2009) - Full Expanded soundtrack