I think the classical argument is that the original rings were Z/nZ, which as you'll remember from “clock arithmetic” has a certain cyclicity to it. This theory is backed up by Wikipedia ( https://en.wikipedia.org/wiki/Ring_(mathematics)#History ) citing Harvey Cohn's number theory book with
According to Harvey Cohn, Hilbert used the term for a specific ring that had the property of "circling directly back" to an element of itself.
(citation from the Wikipedia page as of 26th December 2012)
The Hilbert paper in question is Die Theorie der algebraischen Zahlkörper (Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 4, 1897), credited with introducing the word Zahlring in the first place.
