The civil parish of Errigal Keerogue is the RC parish of Errigal-Kieran. Their baptism records don’t start till 1847, so if that’s where Eliza was baptised in 1842 it doesn't appear there is paper a record to find.
I have searched the parish records on Ancestry and did not find any children to the parents you have given in Co Tyrone. Death registration in Ireland started in 1864. If Patrick & Mary died before that there may be no records to find.
A death in Errigal Keerogue would be registered in Clogher. I searched 1864 to 1901 but did not find a single Patrick. So either he died before 1864 or moved away from that area. There were numerous Mary O’Neill deaths of women of the right age but none had anything to indicate they were married to a husband called Patrick.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee. The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price.
http://www.nifhs.org (Go to DNA project on the website).
