There are many similar definitions of this word that apply to audio and/or video production. In general, the relevant meaning centers around it being a prompt or reminder of some event. It can be traced all the way back to the old days of theatre where actors received or gave cues to trigger other related events on stage. In modern film making a cue is a chunk of music written and performed to go with a specific scene or segment of a film (it is also sometimes called a “start”). The verb, “to cue”, references the positioning of a tape recording at the start of a specific section or a record stylus at a specific cut. Of course these early meanings have now been adapted and modified to correspond to our modern day production techniques.
Also the signal sent to the musicians in a studio or on stage is often called a cue mix and has its origins from the same concepts as above. In order to cue something up, a way to monitor the signal without that signal affecting the main output (whether a radio station signal or audio being recorded to tape) was needed. This became known as the cue mix (for monitoring cues), but is usually called a monitor mix for obvious reasons.