Approximately fifty families first established homes in 1875 near the Fremont River. After residents moved to the townsite from 1895 to 1897 the population increased; it was 327 inhabitants from approximately 120 families in the 1990 census.

Once named Thurber after early settler and explorer Albert King Thurber, the settlement experienced a name change in 1916 when Thomas Bicknell, a prominent educator and historian of Providence, Rhode Island, offered a library of 1,000 books to any town that would take his name. The citizens of Thurber voted to accept the library along with Bicknell's name and recorded the action in April 1916. The town was incorporated in February 1939.

Bicknell lies southwest of Thousand Lake Mountain and gradually slopes toward the Fremont River. The townsite was moved to a higher elevation in 1895 to assure a more adequate water supply. Following the advice of the LDS Church, another site was chosen and surveyed, blocks were laid out, and it was dedicated on 7 June 1895.