Eridu was never powerful. It was small, it sat at the edge of the marshes on the southern rim of the plain, and no empire was ever run from it. What it had instead was priority, and the Sumerians said so at every opportunity. The Sumerian King List begins by stating that kingship came down from heaven at Eridu, and only afterward moves it elsewhere. Literary texts treat Eridu as the oldest place, the point from which the ordered world was set going.
That claim is a claim about Enki, whose house it was. Enki is the god of the abzu, the body of fresh water under the earth, and of everything that requires cleverness: crafts, incantation, the arts of civilisation, and the escape from a difficulty by a device. In the literature he is the god who warns a man to build a boat before a flood he is not supposed to reveal, and the god from whom the arts of civilisation are taken by Inanna at a drinking contest and carried off to Uruk. That last story is a piece of theology about cities: the oldest place is the source, and the newer and larger place holds what came from it.
The atlas holds Eridu and Uruk as a pair for that reason. One is remembered as the beginning and the other was actually the largest city in the world, and Sumerian literature keeps the two in a relationship it clearly enjoyed thinking about.