Hi Ronnie
I was interested to see your message because I have ancestors in that same townland, and some land records which I looked at last year in PRONI in Belfast might be relevant to your search.
I think that the Thomas Martin of your rootsireland transcript, father of a William Martin, was most likely the Thomas Martin who - per information in an 1802 lease - was born in about 1788 to a John "Bane" Martin. That would fit in with the Thomas who died in 1869 as referred to by Elwyn. (Huge kudos to Elwyn and Shane btw for putting up so much information so quickly!)
John Bane Martin took land on lease at Crampan (which seems to have been a sort of sub-townland of Curraghamone/Curraghamongan) in 1802, and I think that's likely to be where Thomas lived and brought up his children. The land shown for Thomas Martin in the late 1850s at Curraghamongan plot 22 in Griffiths Valuation - as mentioned by Elwyn - plus the adjoining land at plot 23, associated with Stephen Martin, Thomas's younger brother (who had died in 1855, according to a Church of Ireland burial record on roots ireland), together seem to correspond almost exactly with the description of the Crampan land in the 1802 lease.
Rootsireland have Church of Ireland baptism records for five, or possibly six, children born to Thomas and Margaret/Peggy Martin between 1821 and 1837: Jane, 1821; William, 1823; William, 1826; Margaret, 1833; Thomas, 1836; and possibly Rebecca Anne, 1837 (Rebecca Anne's mother's name has been transcribed as Mary but this could be a misreading of "Mgt" - unfortunately rootsireland do not provide images of the original register pages so there is no way to check).
Thomas b 1836 seems to correspond to the Thomas who died in 1900 mentioned by Elwyn, and the Margaret who took out the grant of administration for his estate could potentially have been his sister Margaret.
Strangely enough rootsireland do not seem to have a baptism entry for the John Martin born about 1817 who later emigrated to Ohio as mentioned by Elwyn, though from the details on Ancestry trees I think he must have been a child of this couple. If he was, Thomas and Margaret must have married in or before 1817, which may explain why we cannot see a marriage entry for them: the Church of Ireland marriage and burial records for Stranorlar, which may or may not be where they married (often it would have been the bride's parish, which may have been different), only survive for 1821 onwards, though as Shane points out the (C of I) baptism records for the parish go back as far as 1802.
Rootsireland do also have Church of Ireland baptism entries for the children of Thomas's brother Stephen and his wife Hannah.
I think that the James Martin who died in 1880 came from a different branch of the Curraghamone/Curraghamongan Martins. His land was in the western part of the townland centring on Trankeel, where he had a couple of flax mills. James Bane/Bawn/Ban Martin ("Barr" is possibly a transcription error?), who died in 1856, may also have been from the Trankeel branch, though I am not sure about that. (In fact I am not certain whether there was only one, or possibly two, James Banes. Working out which Martins were which is not too easy, since there is not too much information in the records, and there were a lot of Martins in the townland in the late eighteenth and early nineteenth centuries (including any number of James Martins!). They would have been very surprised, I'm sure, to think that by 1901 there would only be two elderly bachelor Martins left to fill in a census return!).
Having said all that - I have to admit that an alarm bell went off for me when you said that the William you are looking for was born in 1823. That's because I don't honestly have the feeling that there was more than one Thomas and Margaret couple in the townland in the 1820s (though of course it's not impossible), and, like Shane, I'd tend to interpret the 1826 baptism of another William Martin son of Thomas and Margaret in the same townland as suggesting that the first William had very likely died in infancy.
Is it possible that your GG grandfather William was born in 1826 rather than 1823? Or, if you are confident that he was born in 1823 but less sure about the connection to Stranorlar parish, you might want to pause before investing too much time and money in investigating the Curraghamone Martins - is it possible that he might actually have come from somewhere else?
It could be worth taking a look at John Grenham's website - sorry I haven't worked out how to paste the link on here, but his site is called johngrenham.com - where you can do lookups by surname to get an idea of where various surnames were geographically concentrated in different parts of Ireland, per various different record sets. He also has sections for individual parishes which show what church records have survived for the various denominations and for what dates, and where any surviving records can be accessed. Sadly the truth is that the records which survive for the early nineteenth century often represent quite a small percentage of the actual baptisms and marriages from that time, so tracing people can be very challenging. But we never give up hope!
Barbara J