Lorna,
The RC baptism records for Dromore list a Brigid Murat baptised 2nd July 1841. No parents are named but given the rarity of the name, it’s possible this child was part of your family. I don't see any other baptisms or marriages.
There’s a possible death listed for Ellen Muratt regd in Omagh on 1st June 1873. That certificate should tell you whether Owen/Eugene was still alive. I can’t see a death for him in the statutory records (which start in 1864) so he may have died before that. You can view Ellen's death on the GRONI site (details below. Use the search registrations option. There is no need to purchaseor order the certificate). That certificate is not available on-line free yet so you need to pay to view it.
Mary Murat’s marriage in 1868 gives her townland (address) as Letteree (more commonly Lettery).
Jacob is the Latin for James. (Many church registers were compilied in Latin). Owen’s son James Muratt married Catherine Breen. They were still alive and living in Lettery in the 1901 census :
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Greenan/Lettery/1741929/
Catherine died in 1904 aged 78. James had died before her but again I can’t find his death certificate.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1904/05628/4583240.pdf
https://geni.nidirect.gov.uk
Griffiths Valuation for 1860 lists Owen Muratt in Lettery. He had plot 15d which was a farm labourer’s cottage. His name remains against that property until 1901 in the Valuation Revision records on the PRONI website, though I suspect he died long before that. His son had evidently acquired the property and the records probably hadn’t been updated.
http://www.askaboutireland.ie/griffith-valuation/index.xml