I know your message was posted some 10 years ago but this might be of some help.
I specialise in Hughes surnames in North County Louth but have not come across a place called Edenamo nor the family to which you refer.
I would say that the family were probably from the Church of Ireland
I have looked up the Hughes name on the Mormon International Genealogical Index and have found
Joseph Parker Hughes born 12 January 1870 in County Louth - no place is given.
His father was William Alexander Hughes and his mother was Mary Parker Hughes.
Does this help?
You may care to obtain a copy of the birth certificate from the Registrar of BMD at Dublin Road Dundalk Co Louth Eire which should give more information. From this you should be able to get the actual place of birth and deduce the parish where you may be able to unearth other members of the family.
Do let me know how you get on as I have over 100 Hughes pedigrees on my file for the Cooley area of County Louth from 1800 to 1900.
Michael Hughes
Buckinghamshire
England