Hi Barry,
Thank you for your message.
Unfortunately, it may be difficult to find information without a more specific place of origin. Most Irish record sources- church records, land records, census records- are based on specific locations. Until you know a specific location within Ireland, preferably a parish or placename, it may be difficult to do anything more with these records.
Do you have any documentation pertaining to the family that mentions any place names at all? This can include wills, marriage/death records,census records, letters, diary entries etc.
There are some places where you can search online, however it may be difficult to discern whether the results are relevant to your family or not without more information:
www.familysearch.org has a huge database of genealogical records including some church records for Ireland.
www.rootsireland.ie have a large collection of Irish Church records, however you have to pay to use this site.
Have you tried looking into the family?s emigration? Sometimes more infomation can be found at the port of arrival rather than that of departure. You may be lucky and find a place name here. For information on US immigration records check http://www.archives.gov/research/immigration/
The surname Eames was not very common in Antrim in mid-19th century Ireland:
http://www.irishtimes.com/ancestor/surname/index.cfm?fuseaction=Go.&Surname=Eames&UserID=
There was only one household recorded, in Island Magee:
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNameSearch&Submit.x=39&Submit.y=22&Submit=Submit&familyname=eames&firstname=&baronyname=&countyname=ANTRIM&unionname=&parishname=
Although it is impossible to know if this Eames is related to Margaret it may act as a starting point if all else fails.
Some other websites that you may find useful are:
The National Archives of Ireland http://www.nationalarchives.ie/genealogy1/genealogy-records/introduction/
The National Library of Ireland http://www.nli.ie/en/family-history-introduction.aspx
The National Archives UK ? genealogy search: http://www.nationalarchives.gov.uk/records/looking-for-person/
The Public Records Office of Northern Ireland http://www.proni.gov.uk/index/family_history.htm
Irish Times: http://www.irishtimes.com/ancestor/index.htm
Irish Genealogy Tool kit: http://www.irish-genealogy-toolkit.com/
Genealogy Links: http://www.genealogylinks.net/uk/ireland/antrim/index.html
From Ireland: www.from-ireland.net
Best wishes,
Genealogy Support