preslinc,
The link you have provided to Griffiths takes me to records in Co. Antrim. I doubt you intended that. However I looked at Griffiths Valuation for Ednafirkin (more commonly Edenaferkin) in Co. Monaghan in 1860 as well as at the tithe applotment records for 1834. There was a James Hamilton farming there in both sets of records. There were only about 8 farms in the townland and just the one Hamilton so a connection to your family wouldn't be too surprising.
http://www.askaboutireland.ie/griffith-valuation/index.xml
http://titheapplotmentbooks.nationalarchives.ie/search/tab/home.jsp
There were no Hamiltons in the townland by the time of the 1901 census. So the family had died out or moved away.
http://www.census.nationalarchives.ie/pages/1901/Monaghan/Caddagh/Edena…
As you know Edenaferkin is in the parish of Tullycorbet. If the family was Protestant then they were probably either Church of Ireland or Presbyterian. Tullycorbet Church of Ireland has records from 1796 onwards (with a few gaps) plus there are 2 Presbyterian churches. Cahans has records from 1751 and Clontibret 2nd from 1858. I don’t think any of these records are on-line but there are copies of all in PRONI (the public record office) in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
I see John Hamilton on numerous trees on Ancestry. All seem to have the same date of birth (17.12.1859 ) but some have his mother as Selina Armstrong and others as Jane Cuddy Beatty. Some have his spouse as Elizabeth Anderson and others as Roberta Cornelia Deyo. He also appears to have died twice. Once in PA on 15.4.1929 and then again there on 1.10.1936. Some have his father Thomas as born in Biggar, Scotland and others in Tyrone, Ireland. Busy man. So some care is needed in ensuring you get the correct records.
There’s an erratic feel to the information on some of the trees. I looked at one (Joe’s tree, which I hope is not yours) and it has Thomas born Tyrone, marrying in Lisburn, Co Antrim, having children in Tyrone, then Monaghan, then Tyrone again before dying in Tyrone. All are possible, just unlikely and it smacks of information trawled from the internet without proper verification. In my opinion it’s likely several unconnected families have been mixed up.
An important thing to understand is that not all birth, death and marriage records in Ireland are on-line, and also that a lot of records are missing. So just because someone finds a record around the right period, and it’s the only one on the database it doesn’t mean it’s right. Especially with a common surname like Hamilton which abounds in Ireland & Scotland.
I am little curious about Thomas A Hamilton being born in Biggar, Lanarkshire, c 1825 and then having children in Co Monaghan in 1859. It’s not a big journey and certainly quite possible. However the major migration of Scots to Ireland was in the 1600s. By the 1800s people, including many of Scottish origins, were pouring out of Ireland because of famine, a population explosion, lack of industry, high rents, and political reasons. That doesn’t mean the odd person didn’t arrive then but it would definitely have been against the trend. (Some big Irish estates were very keen on Scots gardeners and gamekeepers, and there were a few like that who came to Ireland then but the numbers were tiny). So I am wary of the information that Thomas A Hamilton moved from Biggar to Monaghan. What’s the evidence for that? My suspicion is that the Biggar family is probably different to the Monaghan one. But I may be wrong.
But, assuming you are confident that your John Beatty Hamilton was born in Tullycorbet in 1859 or thereabouts, I suggest you get someone to check the church records I have detailed in PRONI to see if his baptism can be confirmed. And also who his parents were. A good researcher will also look for siblings which gives extra lines of potential research.