Hi there,
Altough you have provided lots of details, the most important detail is to find out an exact place of origin. You have given Galway, which is at least a very good start, so lets try to work out a bit more from that.
You can check for information about the frequency of the name in the mid-19th century and any other variant spellings of the name here: http://www.irishtimes.com/ancestor/surname/
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
You could also try checking the land records called the Tithe Applotment Books (1823-38) or the later Griffith'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. 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/ or the Church of Latter Day Saints (LDS).
The Tithe Applotment List might be of use to you, or at least interesting for you. These lists constitute the only nationwide survey for the period, and are valuable because the heaviest burden of the tithes to the Established Church, the Church of Ireland, fell on the poorest, for whom few other records survive. The information in the Tithes is quite basic, typically consisting of townland name, landholder's name, area of land and tithes payable. Many Books also record the landlord's name and an assessment of the economic productivity of the land. the tax payable was based on the average price of wheat and oats over the seven years up to 1823, and was levied at a different rate depending on the quality of land. For Parishes where the registers do not begin until after 1850, this information can be useful, as they are often the only surviving early records. They can provide valuable circumstantial evidence, especially where a holding passed from father to son in the period between the Tithe survey and Griffith's Valuation.
Do you know much about their emigration? 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. There are lots of books and documents available about the Irish emigration to NZ. There is a website you could look at: http://www.teara.govt.nz/en/irish/2 for information about the Irish in New Zealand.
You might also find this link useful for sources: http://www.rootsweb.ancestry.com/~fianna/county/galway/gal-1.html
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)
