I don't know if it's changed, but the law USED to be that if you could prove lineage to two generations, you could get Irish citizenship, and subsequently an Irish passport. That means providing (in the case of a grandparent) the birth/marriage/death (if relevant) certificate for the relative, among other documents. It takes some time and money, but eventually you'll get a passport.
I know this to be true, because I did it back in the 1990s. I have dual citizenship in Ireland. And as someone who has spend a lot of years flying to Europe, it REALLY helps to enter Customs & Immigrations through the "EU Passport" line instead of the "Non-EU Passport" line.