Let's see, if we define favorite as "most sung" song throughout the years, then my favorite is Aragorn’s coronation song. I still sing this song to myself a lot, have always done so, since I first saw Return of the King. It's a very appealing melody to me, probably why I still sing (or hum) it all the time.
But my favorite of all the songs characters sing that I love most to listen to is the Misty Mountain song from An Unexpected Journey. I do hum this one quite a bit, but I'm almost always hearing the full orchestral version in my head from the "Over Hill" cue, not the version with lyrics. (I am not a lyrics person, so no surprise there. That is also the cue that goes with my second favorite moment from An Unexpected Journey, so... again, no surprise that that's the version in my head.) But for sheer listening pleasure, I'll take Thorin and the dwarves any and every day.