"words spoken at the end of a speech in a play that are the signal for an answering speech," 1550s, of uncertain origin. By one theory it is a spelling out of Q, the letter, which was used 16c., 17c. in stage plays to indicate actors' entrances and was explained at the time as an abbreviation of Latin quando "when" (from PIE root *kwo-, stem of relative and interrogative pronouns) or a similar Latin adverb. Shakespeare's printed texts have it as both Q and cue. Cue as a name for "the letter Q" is attested from 1755.
Transferred to music by 1880. Figurative sense of "sign or hint to speak or act" is from 1560s. The television reader's cue-card is attested by 1948.
"tail, something hanging down," variant of queue (n.), ultimately from Latin cauda "tail." Meaning "long roll or plait of a wig or hair worn hanging down, a pigtail," is from 1731. Meaning "straight, tapering rod with a small soft pad, used in billiards," is by 1749. Hence cue-ball, the ball struck by the cue, recorded by 1881.