A Tower Controller reported an aircraft which acknowledged a clearance to line up and wait; took off conflicting with an arrival to an intersecting runway.

Narrative

Aircraft X was joining a 3-mile final for Runway 04 and was cleared to land. Aircraft Y called ready for departure on Runway 14; was cleared for takeoff; and given traffic on Aircraft X landing Runway 04. Shortly after Aircraft Y was cleared for takeoff; Aircraft Z called ready for departure and was cleared for takeoff with traffic being issued. Aircraft Y aborted takeoff and I immediately canceled Aircraft Z's takeoff clearance. I then gave Aircraft Y exiting instructions and asked if they needed assistance; which they did not. I then went back to Aircraft Z and asked if they were past the hold short lines; and they stated that they were. I instructed them to LUAW (Line up and wait) on Runway 14; WHICH THEY READ BACK; and then canceled the landing clearance for Aircraft A on a 3-mile final to the same Runway; I also issued traffic to Aircraft X on about a 1 mile final for Runway 04. During my scan of the Runways; I noticed Aircraft Z was departing Runway 14 and was airborne between A3 and A4 taxiways. I issued a go-around to Aircraft X and then went back and issued traffic to Aircraft Z on Aircraft X going around on Runway 04 to which he responded; 'traffic in sight'. Aircraft X was climbing and I issued Aircraft Z an immediate left turn for the traffic; however Aircraft Z chose to descend to go under Aircraft X. The Tower had both aircraft in sight the entire time. I have no recommendations for this because all appropriate phraseology was used and rules were followed. The pilot was told to LUAW and read back 'line up and wait'. I believe this is a mistake on the part of the pilot.