'Delargy' in Gaelic , spelled slightly different of course (O Duilearga) means 'people from the dark hills ' or 'dark people from the hills'. Some people continue to use that spelling today. The name we are familiar with (Delargy) is the anglicised version which came into being at a time when the English language was beginning to replace the native Irish.
The name originates from Co. Mayo in Ireland. Around the 1500s the Delargy's moved north and settled in the Glens of Antrim and there are still many Delargy's to be found there today. (You can read about this in Lysaght's book of Irish names.)