That's not "Mercury" on the Mercury dime

It's a young Liberty, identifiable by her winged Phrygian cap.

This misattribution appeared almost immediately in the popular press, as writers imagined that the obviously female Liberty was actually a representation of Mercury, messenger to the Roman gods of mythology and quite certainly a male. It is popularly known as the Mercury Dime even today, despite noble but ill-fated attempts by some publications to reverse this error.

From today's featured article at Wikipedia.  And from there we wander to the Phrygian cap:

The Phrygian cap is a soft conical cap with the top pulled forward, associated in antiquity with the inhabitants of Phrygia, a region of central Anatolia. In the western provinces of the Roman Empire it came to signify freedom and the pursuit of liberty, perhaps through a confusion with the pileus, the felt cap of manumitted (emancipated) slaves of ancient Rome. Accordingly, the Phrygian cap is sometimes called a liberty cap; in artistic representations it signifies freedom and the pursuit of liberty...

By wearing the red Phrygian cap the Paris sans-culottes made their Revolutionary ardour and plebeian solidarity immediately recognizable. During the period of the Reign of Terror, the cap was adopted defensively even by those who might be denounced as moderates or aristocrats and were especially keen to advertise their adherence to the new regime... To this day the national emblem of France, Marianne, is shown wearing a Phrygian cap.

The popular comic/cartoon characters The Smurfs are famous for their white Phrygian caps.