Hi,
My surname was originally Ieradi, and was changed to IRADI when my grandfather immigrated to the USA.
The name IRADI is a Basque name, and comes from the name of the region in the Basque Country called "Iraty."
Most Latin Americans with the surname "Iradi" can eventually trace their ancestors back to the Basque Country - the exception being some with the name Iradi who live in and around Buenos Aires, who are the descendants of my grandfather's brother, who had immigrated to Buenos Aires.
The name Iradi in Europe has gained some notoriety, as the leader of the Basque seperatist movement ETA is named Iradi (he is presently in prison in France.)
There is a wonderful ewe's milk cheese sold in Europe called "Iraty-Ossau", which comes from this same section of the Basque country in Spain.
Good Luck with your search!
DMI