Robert Andrews

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. So you probably won’t find statutory birth, death or marriage certificates in Ireland for this family. 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 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

Going by the denominations in the 1901 census of Tyrone and the era when your ancestor left, I’d guess that the Andrews family were Presbyterian (and so probably of Scottish origins). Blythe is mostly a Churhc of Ireland name (ie Episcopalian).

Your task won’t easy because very few parishes in Tyrone have any records for the 1700s. 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).