Hi Sean,
Thank you very much for your message. You have definitely included enough pertinent information! You have obviously done a great job on your family history!
Unfortunately, it may be difficult to find more information on William without a specific parish of origin. Most Irish record sources- church records, land records, census records- are based on specific locations. Until you know a specific location within Ireland, preferably a parish or place name, it may be difficult to do anything more with these records.
Do you have any documentation that mentions any place at all-this can include census records, marriage records, letters, old photographs etc. Sometimes a place of origin can be found on tombstones.
Have you tried looking into the family?s emigration to see if you can find immigration records with a place of origin listed? Here are some links that may be useful when looking into Canadian immigration:
http://www.collectionscanada.gc.ca/index-e.html
https://www.familysearch.org/learn/wiki/en/Ireland_Genealogy_Websites?goback=.gde_3030105_member_230205486#Emigration_Records_.2FPassenger_Lists
http://www.irish-genealogy-toolkit.com/Canadian-immigration-records.html
There are a couple of places online where you can search for records:
www.familysearch.org has a huge database of genealogical records including some church records for Ireland.
www.rootsireland.ie have a large collection of Irish Church records, however you have to pay to use this site.
You could also try looking for record of the families left behind in Griffith?s Valuation. This is a property valuation survey that was carried out in Ireland between 1848 and 1864. It doesn?t have any genealogical information but it can sometimes be used to tie people to specific places. You can search this here:
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameSearch
I did find some Lappins mentioned in the earlier land record called Tithe Applotment Books(1823-1838) for Eglish and Grange parishes in Armagh (check out PRONI website below for information on all parishes), however it is difficult to ascertain whether these Lappins are related to your Lappins:
http://www.connorsgenealogy.com/Armagh/EglishTithes.htm
http://www.connorsgenealogy.net/Armagh/GrangeTithes.htm
There are also Lappans recorded in various parishes on the Flax Growers List. 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.
http://www.failteromhat.com/flax/armagh.htm
Although neither of these links give you anything definitive the parishes mentioned may be places from where to start your research.
Some other websites that may be helpful are:
The National Archives of Ireland http://www.nationalarchives.ie/genealogy1/genealogy-records/introduction/
The National Library of Ireland http://www.nli.ie/en/family-history-introduction.aspx
The National Archives UK ? genealogy search: http://www.nationalarchives.gov.uk/records/looking-for-person/
The Public Records Office of Northern Ireland http://www.proni.gov.uk/index/family_history.htm
Genealogy Links: http://www.genealogylinks.net/uk/ireland/armagh/index.html
The Irish TImes: http://www.irishtimes.com/ancestor/index.htm
Irish Genealogy Tool-kit: http://www.irish-genealogy-toolkit.com/index.html
I hope these are helpful. Remember to post any new information that you find, in particular a parish of origin.
Kind regards,
Genealogy Support