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The Mystery Card
The first known instance of an unknown card set aside magically becoming a card subsequently chosen is Richard Pannwitz's “Der Gipfel der Unmöglichkeit” in Der Zauberspiegel, Vol. 8 No. 4, Jan. 1908, p. 56. Pannwitz employed a duplicate card and some basic sleights to accomplish the trick.
The effect seems to have gone pretty much unnoticed until Theodore Annemann reinvented it, with a clever method that uses a gimmicked card introduced into a borrowed deck. This avoids palming away a duplicate to effect the disappearance of the chosen card from the pack. The gimmicked card became known as "The Nightmare Card".
Annemann didn't give the trick a title when he published it in a two-trick single-sheet monograph called Annemann's Nightmare Effects in 1928. The most commonly used name for the effect, “The Mystery Card”, comes from later approach by Larry Jennings, first published in The West Coast Quarterly, Vol. 1 No. 1, Winter 1984, p. 9; and later in The Classic Magic of Larry Jennings, 1986, p. 34.