Well, not really founders, eponyms rather.
Basically, both characters are credited with giving Ireland his name. Spot the false one!
... Honestly, it is a given that it is the goddess Eriú who gave her name to Ireland. Ireland's native name is "Éire" after all. The guy Ír is mentioned way later but it is interesting that he is basically in the same story as Eriú. And I thought it was interesting that a male eponym existed at all, as male representations of Ireland are traditionally inexistant.
Short summary of the story: Ireland was, according to legends, invaded 5 times. The ancestors of the Irish, named Milesians, were the 5th invasion. (Milesians came from Iberia, their name was based on their founder Míl Espáine, which means "Soldier of Spain") The Tuatha Dé Danann (Child/People of Danu) were the 4th invasion.
Before the invasion of the Milesians, the Tuatha Dé Danann were fighting over something (can't remember what) and it attracted the attention of the Milesians. Concerned, they sent Íth who was their ambassador to Ireland to ease the tension. The Tuatha Dé Danann grew scared over how well Íth was able to express himself with words and so they killed him. That made the Milesians angry and they invaded.
But the Tuatha Dé Danann, being gods and goddesses, used the elements of the island to bring up storms and make the sea very dangerous. One Milesian, named Ír, came with one boat closer to Ireland and managed to set a foot on the island. But the Tuatha Dé Danann used strong winds that shoved Ír over the cliff and back into the sea, subsequently drowning him.
However, other Milesians got nearer to Ireland and managed to land. One of their poets recited an Irish poem and using the power of the words, the storms disappeared and the Tuatha Dé Danann lost their control over the elements of Ireland.
Fighting was quick and bloody. The Milesians and the Tuatha Dé Danann clashed together and many were killed in the struggle. Even Eriú's husband was killed. Later on, Eriú approached the army of Milesians and praised them as the greatest men in the world. Impressed by her words, they asked what she wanted in return. She requested to be named after the island and her wish was granted.
Didn't stop the Milesians from gathering her, her sisters and all the other royal members of the Tuatha Dé Danann together and executing them.
Bet you didn't know that Eriú was married to a man named Mac Gréine, hmm? (his name means "Son of Sun") Also, she was a mother. She had one son named Bres who grew into a tyrannical king. Oh and it is mentioned that she also had an affair. Well, we can't say Eriú was a boring goddess.
Sorry for the huge text, it seemed worthwhile telling you guys about the two characters who are connected by name to Ireland. Ireland doesn't seem overly pleased at having to choose between an executed eponym or a drowned eponym.