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For what it's worth: I cannot find a Margaret (Jane) Harbison marriage in N. Ireland to a Moore so I tried jumping the water to Scotland and found one possibility for you:
Margaret Herbison, her mother's surname was Davidson and she was married in 1856 in Dalry (Ayrshire).  You would have to pay on the SP website to obtain details on who she married. A long shot, I know.

The Margaret Herbison born c. 1830 and daughter of Matthew Herbison and Mary Paul married John REID on 25 June 1859 in 3rd Ballymena Presbyterian church, Ballymena. She died 9 July 1911 in Leymore townland.

Thank you for the comments.  Nancy and Agnes have been well documented as the same person in numerous US and Canadian records. Her nieces/granddaughters, etc. etc. all used the same pattern of naming.

In Canadian records, Martha was nicknamed Matty. Two of her Aunts named their oldest daughters Matilda.  If I assume a traditional naming pattern (cautiously, though), perhaps the maternal Grandmother was "Matilda".

As for Margaret with a nickname or alias as Hannah- that does not seem realistic.

Thank you, Sandra. Unbelievably fast at 12 minutes!   Very much appreciated.   

Thanks, everyone, for the comments.
It is highly possible that James Sr. died between 1861 and 1864 as he would have been in his 70's at the time. So- no civil registration of the death, as you note, Kiltaglassan.
I need to get back to the PRONI website to check out Presbyterian or Church of Ireland records for Armoy to see what's available. But were deaths/burials rarely recorded- if at all?