Your best bet is to start at the public records office here and work back methodically - parents are usually easy. Whoever emigrated will have their age on the death entry so you can estimate the date of birth. You will probably need to know the place of birth in Ireland, because you apply to the local office in that area. I found the Irish registrars to be extremely helpful and as long as you can give them an estimate they will search the records for you and send a copy of the certificates for a few pounds. I was able to track down my Irish grandmother, which gave me the right to Irish citizenship and passport.