"Cingular" isn't one of those completely made-up words used for company names. It's actually an adjective meaning:
  1. Having to do with a cingulum, an anatomical band or girdle on an animal or plant.
  2. Encircling, girdling, surrounding.
and derived from the Latin noun cingulum (girdle), from the verb cingere (to gird). Such words as cincture, precinct, shingles, and succinct come from the same root.

"Differs ... in the greater degree of cingular development on cheek teeth, especially molars."
Daniel L. Gebo, et al; "A Hominoid Genus"; Science (Washington, DC); April 18, 1997.

So presumably the wireless operator chose the name to imply that their service encircled a large area. However, since the word is so obscure and it is pronounced the same as "singular," their meaning is pretty much lost on everyone. (A.Word.A.Day is the only reason I ever heard of the original meaning.)