I can move this thread to Ireland-General board rather than multiple threads looking for the same family.
I think part of the problem is the family information you are started your research with- "came from County Derry, Northern Ireland, at Londonderry on the Foyle River. Bogside near the old historic fort of Londonderry. My grandfather played around the old fort."
Bogside is part of Londonderry (city) near the banks of the Foyle and 'the old fort' could refer to the Walls of Derry. This, if corrrect, points to Derry city rather than the Garvagh area and neither is near Armagh (although the family certainly could have moved).