Perhaps worth copying here the passage about the name from the Wikipedia article on the instrument:

There are many theories for the origin of the name Jew's harp. According to the Oxford English Dictionary, this name appears earliest in Walter Raleigh's Discouerie Guiana in 1596, spelled "Iewes Harp." The "jaw" variant is attested at least as early as 1774[6] and 1809,[7] the "juice" variant appeared only in the late 19th and 20th centuries. It has also been suggested that the name derives from the French "Jeu-trompe" meaning "toy-trumpet".[8]

Theories that the name is a corruption of "jaws" or "jeu" are described by the Oxford English Dictionary as "baseless and inept"; the OED also says that, "More or less satisfactory reasons may be conjectured: e.g. that the instrument was actually made, sold, or imported to England by Jews, or purported to be so; or that it was attributed to them, as a good commercial name, suggesting the trumps and harps mentioned in the Bible