As Wexflyer & Aghadowey have both said, births were sometimes registered years late. I recently came across a family in Belfast where 6 childrens births were all registered together, the eldest being 16 years old at the time. So search widely.
Ultimately you are going to have to satisfy the Irish Passport Office’s requirements as to entitlement to a passport, so it might be worth contacting them to discuss alternative documentation that might be acceptable proof, if you can’t find that key birth certificate. I have known cases in the UK where a sworn affidavit supported by good quality evidence (eg a baptismal certificate) has persuaded the authorities to register a birth late and then you can apply for a passport. But whether that’s the same in Ireland I really cannot say.
I assume you also know that, if someone is claiming Irish citizenship through a father or mother who was not themselves born in Ireland, they need to apply to get that relative's birth registered as an Irish Foreign birth. They have to do that before they can apply for an Irish passport. (The fee is €278). So you’ll also possibly need to persuade them to register it without an Irish birth certificate.
Fuller information on the Irish Embassy website:
https://www.dfa.ie/irish-embassy/great-britain/passports/top-passport-questions/how-much-does-an-irish-passport-cost/