Thanks to Pete Tindal.
Also recorded Oh, Monah – without the apostrophe – in 1941, presumably the same song.
No relation to the Bo Diddley song, Mona, also recorded by The Rolling Stones.
Arnold Rypens’ Originals website cites this as the ultimate source of Ted Weems’s Oh Mo’nah. The recording may be heard at NPR’s Present at the Creation: Barbershop Quartets [link].
The Originals lists several recordings, under various titles, from 1896 until Weems’s as Oh Mo’nah in 1931.