My best guess, and it's only a guess, is that the term "Jew's harp" may well have arisen from an association with the centuries old tradition of itinerant Jewish pack peddlers who commonly sold the small, portable instrument.

I don't believe there were any such pedlars when the word entered the English language - there were no Jews living openly in Britain between the 13th century and when Cromwell let them back in around 1650.