Son of the steward' (irish 'maor'). there were families of this name in armagh, westmeath, offaly and roscommon. the head of the armagh family was hereditary keeper of the book of armagh. mac an mhaoir is also a scottish surname, anglicised macnair, (Sloinnte Gaedheal is Gall )
Sources:
Sloinnte Gaedheal is Gall An Sloinnteoir Gaeilge agus an tAinmneoir