We've known about the 'wet-finger-around-the-wine-glass' idea since Renaissance times—one of the first people to write about that phenomenon was Galileo. Sets of water-tuned glasses on which you can play tunes were popularized in England by Pockridge and Gluck in the early 1700's.
In 1761 Benjamin Franklin was in London representing the Pennsylvania Legislature to Parliament. Franklin was very interested in music: he was a capable amateur musician, attended concerts regularly, and even wrote a string quartet! One of the concerts Franklin attended was by Deleval, a colleague of his in the Royal Academy, who performed on a set of water tuned wineglasses patterned after Pockridge's instrument. Franklin was enchanted, and determined to invent and build 'a more convenient' arrangement.
Franklin's new invention premiered in early 1762, played by Marianne Davies—a well known musician in London who learned to play Franklin's new invention. Initially Franklin named it the 'glassychord', but soon settled on 'armonica' as the name for his new invention—after the Italian word for harmony "armonia". Apparently Franklin built a second instrument for Ms. Davies, as she toured Europe with hers, while Franklin returned to Philadelphia with his own.
The armonica made quite a hit, particularly in Germany. Mozart was introduced to it by Franz Mesmer, who used his to 'mesmerize' his patients, and later Mozart wrote two works for it (a solo armonica piece, and a larger quintet for armonica, flute, oboe, viola and cello). Beethoven also wrote a little piece for amonica and narrator (!), and many of their colleagues of the day composed for it as well—some 200 pieces for armonica (solo, or with other instruments) survive from that era.
But musical fashions changed. Music was moving out of the relatively small aristocratic halls of Mozart's day into the large public concert halls of the 19th century, and without amplification it simply couldn't be heard. During this period, musical instruments in general were significantly redesigned to make them louder to be heard in the larger public concert halls—the piano went through a major transformation from a "quiet little harpsichord with hammers" of Mozart's day to the massive instrument we know today, and instruments of the orchestra—strings, winds, brass—were all modified to increase their volume. But there really wasn't any way to make the armonica louder. Concert reviews from the period bemoan the fact that the armonica sounded wonderful—when it could be heard. So, alas, Franklin's marvelous invention was ultimately abandoned.
Amplification is of course no longer a problem, but even today there are only a dozen or so glass armonica performers worldwide.