"Cue" is derived from the name of the letter "Q", which was used to indicate stage directions in a play. It was an abbreviation for "quando", which means "when" in Latin. "Queue" comes from French for "tail", which comes from Latin "cauda".

http://www.etymonline.com/index.php?term=cue
http://www.etymonline.com/index.php?term=queue

Linguistics and etymology seems to be accepted by /r/askscience as within their purview.