The story
Copper coins minted in Georgian Ireland had the head of the British monarch on the obverse and the Irish harp on the reverse. The phrase "heads or harps" remained common even after 1823, when Irish minting ceased and only British coinage circulated; it was still current when the Irish Free State restored the harp on its newly introduced coins in 1927.
Etymology from Wiktionary, CC BY-SA 4.0.