Can I claim Irish citizenship through a great-grandparent?

The Foreign Births Register route is based on having a grandparent born on the island of Ireland, or a parent who was already an Irish citizen when you were born. A great-grandparent does not, by itself, give you a direct claim.

It can still matter indirectly: if the chain of registrations was kept up in each generation so that your parent was an Irish citizen at your birth, you may qualify through that parent. The key is an unbroken line of timely registrations, not the great-grandparent alone.