Eriu in Irish mythology comes from a Celtic Iveriu, meaning ‘land’, and a further derivative Érainn was used to refer to the bulk of the population of Ireland.
The land-goddess of the country had this (Ériu > Éire) as one of her names, and from it the English and other forms – such as Irlanda – are derived. The dative of Éire is is Éirinn, and its genetive is Éireann.
The goddess Ériu is represented in medieval pseudo-history as one of the three queens of Ireland of the Tuatha de Danann at the time of the invasions by the sons of Mil. She resided at Uisneach(Ushnagh, county westmeath), regarded as the ritual centre of Ireland.
To further underline her function as land-goddess, she is described as wife of the king called Mac Gréine (son of the Sun). The name of this fictional king echos the solar cult of kingship, and the marriage of the male sun with female earth was an archaic and basic convention in Irish ritual.
Extract from: Ó hÓgáin, D. (2006). The Lore of Ireland: An Encyclopaedia of Myth, Legend and Romance. The Boydell Press. pp. 191-192.
Ériu in Irish mythology:
Lebor Gabala Erenn – The Book of Invasions. Part 4.
Some secondary sources for Ériu in Irish mythology:
Bondarenko, G., 2018. Ireland as Mesocosm. In: E. Lyle, ed. Celtic Myth in the 21st Century: The Gods and their Stories in a Global Perspective. Cardiff: University of Wales Press, pp. 53–71
Monaghan, P. 2008. The Encyclopedia of Celtic Mythology and Folklore. New York: Checkmark Books. pp. 160.
Koch, J.T. (ed.) 2006. Celtic Culture: A Historical Encyclopedia. Santa Barbara: ABC-CLIO. pp. 709-717.
MacKillop, J., 2004. A Dictionary of Celtic Mythology. Oxford: Oxford University Press. pp. 169-170