Etymology of PAX proteins
According to InterPro, a paired domain is a DNA-binding element consisting of paired N-terminal and C-terminal subdomains, separated by a linker. PAX proteins follow this motif, and as such typically represent a class of helix-turn-helix transcription factors.

Box is the more interesting term here. In genetics, a box can be used to describe any regulatory element of a gene, including its promoter, UTRs and any enhancer or silencer elements. More loosely they also use box to describe coding segments of a gene, but in older publications.

Going back to Drosophila, however, studies in embryogenesis initially identified a protein called paired, a.k.a. prd. Homology studies into mammalian transcription factors found the presence of a prd-like domain in human HUP transcription factors, and these were renamed to PAX (ref). Thus, the naming of PAX family proteins classicaly denotes the presence not of a PRD regulatory element, but of a coding region for a prd-like domain in the gene product.