I was poring through the Sumerian wordbank when I realized that Sum. eden / edin means both "plain" and "grazing land between two rivers." Could this be the origin of the Hebrew name for their paradise? Recall that a river was said to split into four rivers, and each river fed the land of Eden: the Pishon, Gihon, the Hiddekel, and the Euphrates (Gen. 2.10-14). The Pishon and Gihon have not been positively linked to a real-world river; the Hiddekel is the Tygris and the Euphrates goes by the same name.
Millard (1984) notes that the plain of Babylonia was also called Edin, though Millard argues that the origin of Eden actually lies in the Semitic root 'dn "lush," "abundant." I have little experience in Semitic languages so I will have to defer to Millard's expertise, but the Online Etymology Dictionary sees the origin for the Hebrew word in Akkadian edinnu. I see no reason to not believe that the early Semites were intentionally creating a play on words, a pun of sorts. They contrasted the name of the Sumerian plain Edin with a pre-existing word 'dn via a process called folk etymology.
Millard, A. R. "The Etymology of Eden". Vitas Testamentum. Vol. 34, Is. 1. 1984.