It's called a branchial cleft cyst. The docs use the word "gill" because it's descriptive to lay people since the clefts resemble gills in the embryo. As it turns out, both fish gills AND human parathyroid glands develop from these clefts.
"Branchial pouches" is synonymous with "pharyngeal pouches" which is the term used in this article. "Branchial" = "Of, relating to, or resembling the gills of a fish, their homologous embryonic structures, or the derivatives of their homologous parts in higher animals breathing" (AHED).
Thank you! I knew there had to be a better medical description for this condition.