Most gases are invisible and named for what they do or where they were found. Ozone is named for its smell. The word means, more or less, "the smelly one."
People noticed the smell long before anyone knew what caused it. In 1785 the Dutch scientist Martinus van Marum sparked electricity over water and caught an unusual odor. He put it down to the electrical reactions and never realized he had made a new substance.
About half a century later, the German chemist Christian Friedrich Schönbein noticed the same pungent smell in his experiments. He recognized it as the smell that often follows a bolt of lightning. He isolated the gas and named it after its most obvious feature. Wikipedia dates the naming to 1839; Etymonline gives 1840.
The name comes from the Greek ozon, "smelling," the present participle of ozein, "to smell." Etymonline defines the gas itself as "a modified form of oxygen." It is oxygen with three atoms instead of two. Jacques-Louis Soret worked out the formula O3 in 1865, and Schönbein confirmed it in 1867.
Schönbein had a sense of humor about naming. Others wanted to call ozone "electrified oxygen." In 1844, after showing that phosphorus reacting with air made the same gas, he mocked the idea by proposing "phosphorized oxygen."
The word picked up a popular afterlife in Victorian Britain, where many people believed the bracing smell of the seaside was ozone. Wikipedia notes that this was a folk myth. The smell of the sea comes mostly from dimethyl sulphide made by phytoplankton, not from ozone.
The Greek verb behind ozone has relatives across the Indo-European languages. Etymonline traces them all to a root meaning "to smell." It shows up in Latin as odor, the source of English odor, and as olere, the base of olfactory.