Jill,
Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. For earlier years you usually need to rely on church records, where they exist. Ideally you need to know the precise denomination and have some idea of where the person was born in order to search the correct records. Not all churches have records for that period and not all that do are on-line.
RC records are mostly on-line on the nli site:
https://www.nli.ie/en/family-history-introduction.aspx
For other denominations, the churches usually hold the originals but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
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).
1901 Irish census for Co. Tyrone has 89 people named Arthur and 46 Arthurs. Nobody at all named Pettigrew (or variants). Griffiths Valuation for 1860 lists just a single Pettigrew in Tyrone. He was Charles Pettigrew who lived in the townland of Bohard in Aghaloo parish where he had a 24 acre farm (On the modern Tullybleety Rd). In the absence of any other clues, you could start by searching church records for Aghaloo in PRONI in Belfast.
