Hi there,
Could you please post your message here: http://www.irelandxo.com/group/county-mayo
It is very difficult to carry out research without an exact place of origin or more specific dates. Do you have any other documentation in your family, such as family bibles or other official documents? These may give more clues. You should consider asking other relatives for information. Someone else might know the vital clue that you need to get you started.
I can suggest some websites you could look at, but really you will need more details before you can really get started.
Civil registration records are available from the General Register Office (GRO). These start from 1864 however. You can access the website here: http://www.groireland.ie/research.htm
Have you tried checking the Ellis Island records (www.ellisisland.org) or the Castle Garden records (www.castlegarden.org)? Generally, more information was given at the port of arrival rather than the port of departure. If you knew which city they arrived at, this could be a good place to find more information, and perhaps even find out an exact place of origin.
You could also try checking the land records called the Tithe Applotment Books (1823-38) or the later Griffth's Valuation (1848-64). Griffith's is freely available here: www.askaboutireland.com or here: www.failteromhat.com Failte Romhat has lots of other useful links you could try looking at.
You could also take a look at the 1901/1911 census records to check for the frequency of the surnames in those counties. It might help to give you some leads perhaps. You can check this here: http://www.census.nationalarchives.ie/search/
Please make sure you link anyone else in your family who is interested in their Irish heritage to our site - and indeed anyone else you know of Irish heritage.
Kind regards,
Sinead Cooney
Genealogist (Ireland XO)