TOD:
I'm certainly not as well-versed as others on musical theater, but I never hear any George M. Cohan except when I watch YANKEE DOODLE DANDY. Maybe those songs are not well-travelled over the years, I'm not sure.
Not to duplicate what others have said, and to perhaps reveal my lack of education in this subject, I would start with the lyricists, because I personally have no trouble getting the tune on first listen, it's the lyrics that don't always make intuitive sense to me. But, the two brothers whose lyrics for me always became as memorable as the tune, are Hal and Mack David. They co-wrote with many great composers.
Following from Mack David, how about one of his his frequent collaborators, Jerry Livingston? These I think would be fun to hear on stage. F'rinstance, there could be a medley of the many TV themes which would undoubtedly be a rich source of in-between patter. Who wouldn't love to hear the theme from "Mister Ed" done in an ironic way like a serious classical song? Plus, the other fun songs like "Bibbidi Bobbidi Boo" and "Mairzy Doats", and "The Bugs Bunny Theme", too.
These are my two cents, and maybe they're not even worth that, but it's a post, and I chimed in, and it's my story, and I'm sticking with it, and that's the truth [cue Lily Tomlin and her Bronx cheer].
