I vote Melody for #1 and Rhythm #2. Anyone who has ever done solo guitar gigs will probably understand this. You can leave out harmony altogether and people can recognize songs by the melody alone, providing you play it in time.
Melody is to a song like words are to a poem. The melody is what makes a musical statement. A strong, recognizable melody will make all your original compositions memorable. The opposite is also true of weak melodies. (I can name that tune in three notes …) The melody is the signature of a song, its ultimate identity. I guess that’s why I like older music (Gershwin, Cole Porter, Kern, Hart, Mercer, Mancini, etc.) more than contemporary music. The melodies are so much better.
How many of you really pay as much attention to developing good, strong melodies as you should? The pinnacle for me is being able to play a smooth chord-melody in time. When I open up the Real Book and practice tunes like Autumn Leaves for instance, I always try to play the melody correctly at least once in the opening verse. After that, my soloing can point back towards the melody and even resolve back into the melody if I like.
A song without a good melody is like a burger without the beef; like beer with no alcohol; like fries without ketchup; like a day without sunshine; like a (fill in the blank). I’m not trying to say that harmony and rhythm aren’t also important, but if you’re going to blow something, please don’t let it be the melody.