"Owyhee" and "Hawaii" are two different spellings for the same word. When Captain James Cook discovered what he named the Sandwich Islands (known more recently as the Hawaiian Islands) in 1778, he found them inhabited by people called Owyhees. The spelling "Owyhee" is simplified a little from its original form: "Owyhee" is the spelling that British and American traders used during the early nineteenth century in referring to natives of the Sandwich Islands, and a number of Owyhees sailed on to the Columbia, where they joined trapping expeditions or worked at some of the fur trade posts.
Three of the Owyhees joined Donald MacKenzie's Snake expedition, which went out annually into the Snake country for the North West Company--a Montreal organization of Canadian fur traders. Unluckily, those three Owyhees left the main party during the winter of 1819-20; they set out to explore the then unknown terrain of what since has been called the Owyhee river and mountains, and have not been heard from since. Because of their disappearance, the British fur trappers started to call the region "Owyhee," and the name stuck.
Just at the time the Owyhees disappeared into the Owyhee country, American missionaries came to the Sandwich Islands and worked out an alphabet for the native language in order to print the Bible and other missionary literature. In the alphabet they adopted, the word "Owyhee" turn out to be "Hawaii."
But in Idaho, the older form survived. Many of the fur traders' Idaho place names were lost in later years, but some--including "Owyhee" for a mountain range and river--were retained. That may result in part from the fact that Owyhees remained active in the Idaho fur trade right down to the last years of its decline: as late as 1850, Fort Boise (located on the Snake River just below the mouth of the Owyhee) was staffed by James Craggie and fourteen Owyhees. When the Owyhee mines were discovered in 1863, the name still was in use. And the mines brought permanent settlement which preserved the name ever since that time.