From Maximilian Rudwin’s The Devil in Legend and Literature [1931], pp.256-7:
"While the Devil plays all instruments equally well, he seems to prefer the violin. He was said in the Middle Ages to own a violin with which he could set whole cities… to dancing… until they fell dead… The Devil appears in this rôle in the medieval legend of the Pied Piper…
“Satan appears as a fiddler in the poem ‘Der Teuful mit der Geige’ … of the 16th century.”
Rudwin then lists a bunch of 19th-century literary devil/fiddle references to things like the opera Un Violon du Diable (1849) and Benjamin Webster’s “The Devil’s Violin” (1849).
I think Clairobscur has it in general; the fiddle just happened to be one of the possibilities that made the leap from an idea in folk culture to Western popular culture in the 19th century, when the image was fixed.