Antarctica is named for a bear that has never lived there. In fact, no bear lives there at all.
The bear is in the sky. The Greek word arktos meant "bear," and it also named the constellation we now call Ursa Major, the Great Bear, which circles the northern sky. From it came arktikos, "of the Bear," and so "of the north." Etymonline gives this as the root of Arctic.
Add the Greek prefix anti-, "opposite," and you get antarktikos, "opposite the north." That is the whole trick. The south is named only by pointing at the north and saying "the other one."
The word is old. Wikipedia reports that Aristotle wrote about an "Antarctic region" around 350 BC. It passed through Latin antarcticus and Old French antartique into Middle English. Etymonline dates the English adjective and its noun use for the southern polar regions to the late 14th century.
For centuries it described an imagined place. Europeans pictured a great southern landmass to balance the northern continents. It was called Terra Australis, and no one had seen it. Wikipedia adds that when Sydney's authorities renamed New Holland as Australia in 1824, "Terra Australis" was no longer available for the real ice continent.
Europeans probably first sighted the continent in 1820, during the expedition led by Bellingshausen and Lazarev, according to Etymonline. But the continent still lacked a settled name.
The National Library of Scotland credits John George Bartholomew, an Edinburgh mapmaker, with the first map to label it "Antarctica." His 1887 South Polar chart for the Handy Reference Atlas placed the word above the older description "Unexplored South Polar Continent." Wikipedia says other names such as Ultima and Antipodea were suggested. The library says Antarctica was firmly established by the end of the 1920s.
One small spelling oddity remains. Etymonline says the "c" was dropped in Medieval Latin and Old French, as in antartique, and English put it back in the 17th century.