"bad smell," 1620s, probably from the verb funk in the sense "blow smoke upon; stifle with offensive vapor" (though this is not recorded until later 17c.). It is from dialectal French funkière "to smoke," from Old French fungier "give off smoke; fill with smoke," from Latin fumigare "to smoke" (see fume (n.)).
Not considered to be related to obsolete funk (n.) "a spark," mid-14c., fonke, a general Germanic word (compare Dutch vonk, Old High German funcho, German Funke. The Middle English word is probably from Low German or from an unrecorded Old English form.
In reference to a style of music felt to have a strong, earthy quality, it is attested by 1959, a back-formation from funky (q.v.).