According to oral history of the Edo, Igodomigodo was the name of the kingdom as given by Igodo, the first ogiso (King). The ogiso era, started by Igodo, would be the first dynasty of what would later be known as the Benin Empire (which itself existed from around 1180 until 1897 in the area of the present-day Federal Republic of Nigeria (and remarkably not in the area of the unrelated, current Republic of Benin).
Replacing the era of the ogiso, the oba dynasty was founded by Eweka I, the first Oba (and grandchild of the last ogiso). Eweka I renamed his empire Edo, and its people have been since then called Edo people (or ovi-edo, which means "childr(en) of Edo").
The current Oba, Erediauwa I, is the 39th Oba of the dynasty ultimately started by Igodo of Igodomigodo.