Green’s Dictionary of Slang

a.o.h. n.

also ancient order
[abbr.; ancient order of hibernia]

(US short order) a potato.

[US]Boston Globe 24 Feb. 18: Potatoes had as many names as a Spanish princess or a bank cashier in Canada. Here are a few of them: Spuds, Murphies, T.D.’s, ancient orders or A.O.H.’s and St. Clairs.