I once spent a lot of time trying to find the origin of the term "Jew's Harp"
without much success. The term appears to be very old. I suspect that
"juice harp" is a relatively recent variant, possibly an effort to find a less
offensive name, but that's just my guess. I did not see it in any early
sources. Here are a couple of citations that may be useful:
The New Grove Dictionary of Musical Instruments (a standard work in
music research) says that "The origin of the name remains obscure" and
notes that the instrument has never been associated with the Jewish
people. It also dismisses the suggestion that the term was actually
"jaw's harp." (I'm sure that term has been suggested frequently because
of the fact that the instrument is held between the teeth.) They also note
that the instrument has been known in Europe "at least since
Gallo-Roman times." The Oxford English Dictionary cites printed
references to the instrument from the late 16th century onward,
beginning with Hakluyt's Voyages (1596) which mentions "Iewes
harpes."
I hope this helps a bit.
Kathryn Kalmanson