The Answer:

The word "Hoosier" has been associated with the state of Indiana and its inhabitants since the 1830s. Although the origin of the word is still up for debate, there is no shortage of theories.

According to the Indiana Historical Bureau, the term gained popularity in the state after the Indianapolis Journal published "The Hoosier's Nest"—a poem by John Finley, of Virginia—in 1833. Written uses of the word date back at least as far as 1827; how long it was used orally before then is unknown.

One popular theory about the word's origin is that when Indiana's first settlers responded to a knock on their cabin door they would say "Who's yere?" And from that Indiana became the "Who's yere State" or "Hoosier State."

The Indiana Historical Bureau states that

...by all odds the most serious student of the matter was Jacob Piatt Dunn, Jr., Indiana historian and longtime secretary of the Indiana Historical Society. Dunn noted that "hoosier" was frequently used in many parts of the South in the 19th century for woodsmen or rough hill people. He traced the word back to "hoozer," in the Cumberland dialect of England. This derives from the Anglo-Saxon word "hoo" meaning high or hill. In the Cumberland dialect, the word "hoozer" meant anything unusually large, presumably like a hill. It is not hard to see how this word was attached to a hill dweller or highlander. Immigrants from Cumberland, England, settled in the southern mountains (Cumberland Mountains, Cumberland River, Cumberland Gap, etc.). Their descendents brought the name with them when they settled in the hills of southern Indiana.

—The Fact Monster