Judson Brown invented what is clearly the basis for this classic trick. Brown's inspiration was a prepared deck that could make any card chosen disappear: Eric Impey's “Mysto, the Masterpiece” in his manuscript Original Card Mysteries, 1928. Using Impey's idea of using waxed-together pairs of cards, Brown adapted it to make any card named turn face up in the deck: “A Super-Reverse Problem” in The Sphinx, Vol. 28 No. 1, Mar. 1929, p. 25.
Dai Vernon's “Brain Wave Deck” was originally published in The Jinx, No. 49, Oct. 1938, p. 341. Vernon’s original trick, which he created in 1930, was a technical refinement on Brown's. Vernon replaced wax with roughing fluid to hold the pairs of cards together. Vernon credited Paul Fox with the later idea of adding a kicker; that the reversed card has a contrasting back from the rest of the deck.
These are the essential components of the history of “Brainwave”. However, the story of its evolution is long and complex. In A History of the Brainwave Principle (1983), Karl Fulves offers an excellent and detailed look at how this effect evolved.