Do you know what religion your ancestors were?
There are Roman Catholic church records available for the diocese of Tynan (Armagh). These date from 1822 for both baptismal and marriage records. You can find these at the National Library of Ireland (NLI), Pos. 5589, or the Public Records of Northern Ireland (PRONI), MIC.1D/40, or the Church of Latter Day Saints, ref. no. 09799710, item 6-8. If you have any difficulty, you could try writing to the parish priest for more assistance.
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).
Do you know much about their emigration? Dates, the reason why they 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, this could be a good place to find more information.. 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 find this link useful: http://www.theirisharchives.com/counties/view/20/Armagh
1796: Spinning Wheel Premium Entitlement Lists This was part of a government scheme to encourage the linen trade, free spinning wheels or looms were granted to individuals planting a certain area of land with flax. The lists of those entitled to the awards, covering almost 60,000 individuals, were published in 1796, and record only the names of the individuals and the civil parish in which they lived. The majority, were in Ulster, but some names appear from every county except Dublin and Wicklow. A microfiche index to the lists is available in the National Archives, and in PRONI. You can check www.failteromhat.com for information and lists. There were 3,100 names for Co. Armagh.
1803-31: Armagh Freeholders and Poll Books PRONI ARM 5/2/1-17, D 1928. Also NLI Ir. 94116 a 1, Ir. 352 p 2.
There is a book by J. J. Marshall you might find interesting called 'History of the parish of Tynan in the County of Armagh: with notices of the O'Neill, Hovenden, Stronge and other families connected with the district' (1932).
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.
Genealogist (Ireland XO)