Ó Muireadhaigh - anglicised as Murray, and earlier as (O) Murrihy and Murry, means possibly -seaman/mariner or -lord.
Mac Muireadaigh - anglicised Murray and Murry, and earlier as MacMurray etc. and the same possible derivations as the above name.
Mac Giolla Mhuire - anglicised as Kilmore, Gilmore, Murry and Murray, meaning - son of the servant of Mary.
Ó Muireadhaigh was the name of different septs. Woulfe, in 'Sloinnte Gaedheal is Gall' (1923), says the most powerful were in counties Roscommon (of the Uí Maine), Mayo, Westmeath and Cork. They were also found in Donegal and Cavan. The families are now widespread and dispersed. The chief of the Uí Maine sept appears in the 'Composition of Connacht' in 1585.
Mac Muireadhaigh is of Co Donegal, and of Co Down, according to MacLysaght ('Irish Families' series 1980s) anciently in that county. It has been suggested, too, that families of (Mac)Murray of east Ulster may come from a family of that name in Galloway, south-west Scotland; others in Ulster could be settlers from the Scottish Highland Clan Murray also.
Woulfe adds that there was an old native Breffney (Leitrim) family of this name, who are recorded early in the 16th century 'Composition Book of Connacht'.
By the time of the 1659 Census of Sir William Petty, the following are relevant under the description of 'Principal Irish Name':
Westmeath, Rathconrath Barony, Murrey/Morry (12 families); Delvin Barony, Murrey (8); Roscommon, Ballintobber Bar. Murrey (8); Athlone Bar. O Morey/Murry (32);
Down, Castlereagh Bar. Mc Murry and O Murry (20)
Cork, Kilnameaky Bar. O Morahow (11)
In Griffith's 'Primary Valuation' (1848-60) the counties with the highest returns for Murray were: Cork (356) Roscommon (278) Down (277) Galway (262) and Westmeath (224)
The Registrar General's Report on Births in 1890 listed most in Dublin, Antrim, Cork, Down, Galway and Mayo.
Two Famous Irish Murrays:
Donogh O' Murry was bishop of Athlone in the mid 15th century.
Paddy Murray, formerly of the Provisional IRA, and now a dissident Republican