Mcmahon — Son of the Bear

McMahon renders Mac Mathghamhna, ‘son of Mathghamhain’ — the bear. The name arose twice and quite separately: among the Dál gCais of west Clare, and again in Monaghan, where the MacMahons ruled the kingdom of Oriel from the thirteenth century to the sixteenth, out of the barony of Truagh. Mathghamhain was borne by a brother of Brian Boru.

Bears were long gone from Ireland before either family took the name — which tells you it was the idea of the animal they wanted, not the animal.

Irish form: Mac Mathghamhna