English borrowed this potentially useful word from French about two centuries ago, though it has long since abandoned it again. A search of newspaper archives suggests that it’s used nowadays merely as a rare word with which to stump contestants in US spelling bees.
The French continue to use it, hyphenated, for the bird that we call a flycatcher, appropriately so since it is made up of gober, to swallow, and mouche, a fly. In French it also means a credulous person who accepts everything said to him as the plain truth.
Only the latter sense came over into English:
These people are great gobemouches; they always report the most incredible things.
Travels in the Great Desert of Sahara, in the Years of 1845 and 1846, by James Richardson, 1848.
The inescapable image is of a naive individual thunderstruck by the world around him, perpetually open-mouthed in astonishment and ready to swallow whatever came his way, whether flies or tall tales. This sense of the word is said to have been popularised in French through a play of 1759 by Charles Favart, La Soirée des Boulevards, which featured a character named Gobemouche.
It’s tempting to see a connection between gobemouche and gob, that infelicitous monosyllable which has been a British dialect and slang term for the mouth since the sixteenth century. The latter is most likely from the Gaelic and Irish word for a beak or mouth; if so, then there’s indeed a link, as the French gober originated in the related pre-Roman Celtic tongue called Gaulish.