Cotton says that “the game is so called from the silence that is to be observed in the play.” Dr. Johnson has adopted this derivation; but Taylor the Water-poet (1650), Swift (1728), and Barrington (1787) called the game Whisk, to the great discomfiture of this etymology. Pope (1715) called it whist.
The first known mention of whist in print was in a book called The Motto, published in 1621, where it is called whisk. The earliest known use of the present spelling is in Butler's Hudibras (1663).
Let nice Piquette the boast of France remain, And studious Ombre be the pride of Spain; Invention's praise shall England yield to none, While she can call delightful Whist her own.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894