James Emmet Trainor’s father was Christopher. His will is on the PRONI wills website. Dated 1883, It mentions children James, Alexander & Catherine, also a sister in law Catherine Donnelly.
Tyrone Courier of 8th Dec 1888 has a notice regarding a civil court case involving Alexander Trainor. It appears that John Dillon owed Alexander money. Alexander took him to court and obtained an order relating to just under 6 acres of land in Farsnagh, owned by Dillon. The land was to be sold to pay off the debt.
Catherine Trainor death in 1918:
https://www.irishgenealogy.ie/files/civil/deaths_returns/deaths_1918/05189/4433398.pdf
Looks as though there was a Terence Treanor who was a brother to Catherine:
Administration of the Estate of the late Catherine Treanor, Farsnagh, Co. Tyrone, Spinster, who died 19 June 1918, granted at Armagh 24 May 1919 to Terence Treanor, Farmer, the brother. Effects: £34 1s 3d.
Mid Ulster Mail of 18th Nov 1922 reports that at Coagh Petty Sessions William J Trainor of Farsnagh was fined 2 shillings and sixpence for driving a cart with no lights at night.
Mid Ulster Mail of 5th June 1954 mentions a Mrs Elizabeth Trainor of Farsnagh winning £2 in the local Parochial prize draw. So evidently there were still Trainors there then. Possible death for Elizabeth 28.8.1970 aged 75. You can view that on the GRONI website. You might want to knock on a few doors at Ardboe Cross to see if the name is still known in the area. As far as I can see from Google Earth the pub and shop are gone but there is a farm or former farm opposite, on the other side of the road.
The farm was probably known by the name of “Old Cross House”. I see a notice in the mid Ulster Mail of 2nd October 1943 in which Mrs E Trainor of that address announced that the sale of the farm and its lands had been postponed. She was still there in 1959 because she was advertising fields to rent for crop growing and grazing.