My aunt used to sing this to me when we'd go walking at night. I was very young, so I only remember a rudimentary set of lyrics, but her version seemed to differ even more than the above offerings:

Oh Mister Moon, Moon
Bright and Silv'ry Moon,
Won't you please shine down on me?

[repeat]

Oh when your life's in danger, and you're 'fraid to run, and there's a man behind you with a big shotgun...

Oh Mister Moon, Moon
Bright and Silv'ry Moon,
Won't you please shine down on me?

Doo-do-doo-do-doo-doo,

Please...shine...down on........ME!


Like I said, incomplete at best, but that's why I'm out looking now!