Wikipedia has some ideas, or dare I say, speculations:
The precise origin of the phrases "bull market" and "bear market" are obscure. The Oxford English Dictionary cites an 1891 use of the term "bull market". In French "bulle spéculative" refers to a speculative market bubble. The Online Etymology Dictionary relates the word "bull" to "inflate, swell", and dates its stock market connotation to 1714.
The fighting styles of both animals may have a major impact on the names. When a bull fights it swipes its horns up; when a bear fights it swipes down on its opponents with its paws. When the market is going up, it is similar to a bull swiping up with its horns. When the market is going down it is similar to a bear swinging its paws down.
One hypothetical etymology points to London bearskin "jobbers" (market makers), who would sell bearskins before the bears had actually been caught in contradiction of the proverb ne vendez pas la peau de l'ours avant de l’avoir tué ("don't sell the bearskin before you've killed the bear")—an admonition against over-optimism. By the time of the South Sea Bubble of 1721, the bear was also associated with short selling; jobbers would sell bearskins they did not own in anticipation of falling prices, which would enable them to buy them later for an additional profit.
Another plausible origin is from the word "bulla" which means bill, or contract. When a market is rising, holders of contracts for future delivery of a commodity see the value of their contract increase. However in a falling market, the counterparties—the "bearers" of the commodity to be delivered—win because they have locked in a future delivery price that is higher than the current price.