HI Mary,
It may be better to begin with Michael's immigration. Do you know much about his emigration? Dates, the reason why he left, etc.? Generally more information was given at the port of arrival rather than the port of departure. If you knew which city they arrived at (Castle Garden, Ellis Island), this could be a good place to find more information, and perhaps even find out an exact place of origin. There are free searchable sites for this at http://www.castlegarden.org/ and http://www.ellisisland.org/ . Often shipping manifests can also be checked.
Another possible source of information may be https://familysearch.org/ which is a free site provided by the Church of Latter Day Saints. This site often has census returns, birth & death notices and other documents related to Irish immigrants and may be a good place to look for Michael.
Also when looking at records it is important to be aware of the variety of spellings that may have been used at different times. You can check for information about the frequency of the name in Ireland in the mid-19th century, the various counties the name was accociated with and any other variant spellings of the name here: http://www.irishtimes.com/ancestor/surname/ .
If a place of origin could be established, i.e. parish, county or town, you could try checking the land records called the Tithe Applotment Books (1823-38) or the later Griffith's Valuation (1848-64). The Tithe Applotment Books (1823-38): Microfilm copies of the books for all of Ireland are available at the National Archives of Ireland (NAI) http://www.nationalarchives.ie/genealogy1/genealogy-records/tithe-applotment-books-and-the-primary-griffith-valuation/ . 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. Also the Valuations office in Dublin (http://www.valoff.ie) will have a record of the land registry from 1855 to modern times. This will assist in seeing what happened to Michael and any land he may have owned (as it usually passed on to a relation).
You could also try the National Archives at http://www.nationalarchives.ie/ for information on possible church records.
I wish you the best of luck in your 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,
Genealogy Support
Ireland Reaching Out
