What is the reasoning behind naming proteins first found in Drosophila as paired box? All I could find on internet is that it was first found in Drosophila as a protein with paired domain, but I couldn't find anything about the term box along with it. Could anyone please signify the terms paired and box in the name of that protein?
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.
In addition to the other answer, the following is some historical context on the naming of these genes.
1980 October: Discovery of pair-rule genes in Drosophila.
We have undertaken a systematic search for mutations that affect the segmental pattern depending on the zygotic genome... In [pair-rule mutants,] homologous parts of the pattern are deleted in every other segment
1986 May: Isolation of the paired gene in Drosophila.
Here we report the isolation of the pair-rule gene paired (prd)... Transcripts of prd exhibit a pattern of seven evenly spaced bands during late syncytial blastoderm, reflecting a double-segment periodicity...
1986 December: Characterization of the paired box domain in paired.
...we show here that the prd gene contains a third domain about twice the size of the homeo domain, the paired box...
1988 May: First use of the abbreviation Pax to describe paired box containing murine homologs to paired.
With the objective of isolating murine paired box homologs, a mouse paired box sequence (Pax 1) was identified that shows extensive nucleotide and amino acid sequence homology to all three known Drosophila paired box genes.
In response to a comment on the other answer:
Paired box refers to a DNA sequence/motif. The actual paired domain is a DNA binding motif present in some transcription factors. The crystal structure of the paired domain from prd has been solved. PDB: 1PDN. Expectedly, it doesn't really look like as box.