Doug,
Birth registration started in Ireland in 1864. I searched the GRONI indexes for children born to James & Matilda 1864 onwards and found:
James 8.1.1864
Robert 23.9.1866 Aughnamoe
Jane 19.9.1868
Sarah 22.1.1871 Aughnamoe
Martha 23.5.1873
Catherine 22.10.1875
Alicea 10.11.1878
For children born before 1864 you would need to search the local church records, which are likely to be in PRONI. Dromore Presbyterian’s baptisms start in 1835. Matilda was born c 1838 so you may find her baptism there too, together with siblings if there were any.
Matilda died 5.4.1888 aged 50.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_retu…
James and 2 children in 1901 in Aughnamoe:
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Tullyclunagh/Aughnamoe/1741978/
James in 1911:
http://www.census.nationalarchives.ie/pages/1911/Tyrone/Tullyclunagh/Aghnamoe/870684/
Probate abstract:
Probate of the Will of James Patterson late of Aughnamoe, Dromore, Co. Tyrone, retired farmer, who died 16 December 1917 granted at Londonderry to James Patterson, Aughnamoe, Dromore, farmer. Effects: £61 2s 6d.
The will itself is not on-line. It should be in PRONI in paper format.
There was a second Patterson farm in Aughnamoe. Presumably the 2 were related.