A death record is a good place to begin searching an Irish-born ancestor. If he or she died in the U.S. then the amount and accuracy of the information on a death certificate will vary from location to location, time period to time period and certificate to certificate.
Some death certificates will identify the Irish county of origin for the deceased. A smaller set might list even the city or parish of origin in Ireland. Many will list only Ireland.
Other information on the death certificate might include when the deceased came to the USA and how long he or she had resided in the current municipality. The names of his parents are often listed, including mother's maiden name.
One must be careful when gleaning information from a death certificate. Oftentimes, the information may include mistakes. One record I have lists the widow's maiden name as the maiden name of the deceased's mother! On this same certificate the deceased's father's name is also incorrect, so one could wind up on a wild goose chase.
Another good resource created at the time of death is the obituary. In rural areas obituaries generally ran page one and can provide a wealth of information, including place of origin.
The United States Census can be a tool for determining the Irish origins of our immigrant ancestors. Taken every ten years and becoming more and more accessible to searching online, the U.S. Census rarely identifies locations in Ireland. However, the 1930 Census does list places of birth for some people (or their parents) with terms like Irish Free State, Western Ireland etc. While these terms do not give an exact location, they do whittle down the number of 32 county candidates to a section of Ireland.
The U.S. Census also identifies if the ancestor naturalized and when he or she immigrated to America. It will identify who lived with the ancestor throughout the years. If a date of death has not been determined, the census could help narrow the window. If an immigrant couple are enumerated in 1900 but the wife appears as a widow alone in the 1910 Census, her husband most likely died between 1900-1910. The 1900 and 1910 U.S. Census records will list the number of years married and the number of children born and still living to each mother. The 1900 Census lists the month and year of birth for each person enumerated.
In addition to the federal census many states also counted their residents. While the 1890 U.S. Census was mostly destroyed by fire, states like Nebraska and Iowa conducted their own counts in 1885. New York City was unhappy with the 1890 headcount so it conducted a re-count in October of 1890, most of which survives for Manhattan and the Bronx. This police census provides much less information than the 1890 federal census collected. New York state censuses also survive for other years including 1875, 1892, 1905, 1915 and 1925. Some counties have more thorough collections than others.
Iowa conducted many censuses from 1856 to 1885, 1905, 1915 and 1925. The 1925 Iowa Census provided the names of parents for those enumerated. It also listed the religion of Iowans. Be sure to determine which state censuses exist for each locality an ancestor called home.
Naturalization records vary in detail. Those naturalized after 1906 have a much more detailed record of place of origin on their Declaration of Intent and Petition for Citizenship. These records are available through the regional National Archives and will provide place of origin. Earlier records don't usually provide as much detail but could list an exact date of arrival or provide the name of a sponsor who is a relative.
Later passenger lists show which American the passenger is going to see and disclose a contact person in the home country. Even if your ancestor came in 1897, his niece's passenger list from 1914 may reveal the place of origin for both of them in County Cork!
Collateral relatives are a resource to utilize. It is important to identify the entire group of Irish-born siblings and sometimes cousins. As a group it is more likely that the place of origin is identified in various records. A brother who fought in the Civil War may have identified where in Ireland he was from on a pension application. This record may prove priceless decades later. Later Catholic marriage records will often list the baptismal information of the couples, including parents' names and place of baptism. A bachelor brother who finally married at age 50 in 1920 may be the key to revealing the parish of origin in Ireland.
The International Genealogical Index(IGI) is a resource that must not be overlooked. While no man-made index is perfect, this record includes quite a bit of extracted records that warrant the time searching it. Many Manhattan births and marriages are included. In the British Isles section, several years of Irish births are included beginning in 1864. (Civil registration of births in Ireland commenced in 1864, as opposed to 1855 in Scotland and 1837 for England and Wales.) If your ancestors possibly had a child between 1864-1870 or so, it is worth a search in the IGI. Even if an ancestor was born in Ireland in 1850 there is a chance that he or she may have a last sibling born to the same parents in 1864. This child's birth registration will provide the location. One must know the father's name and surname as well as the mother's first name, at a minimum, to perform the search. There are no guarantees but one will have an answer in a matter of seconds.
Lastly, one should consider checking the countries in Great Britain. Many Irish settled in Scotland or England before making the trip to America or elsewhere. If they married, had a child, or were enumerated in a census while in Britain, they may detail where in Ireland they were born. Many of these resources are available online.
Irish genealogy is considered a challenge by most. It should not intimidate anyone but inspire the researcher to knuckle under and search under every stone.