A Tower Controller reported a departing aircraft deviated from the SID and flew below the minimum vectoring altitude.

Narrative

I had a sequence of departing Runway XXR at [Taxiway] 1 with a ZZZZZ1 (right turn); loading Runway XXR at [Taxiway] 1 with a ZZZZZ2 (straight out) and interspersed a ZZZZZ3 Runway XYL that would go behind my ZZZZZ1. I also had a transition coming in from the northwest that would pass well behind the two XXR departures. The ZZZZZ1 was cleared for takeoff. The ZZZZZ2 was cleared to line up and wait as well as advised the aircraft ahead would be a right crosswind departure as initial separation. I then exited a Runway XXL arrival; crossed Runway XXL at [Taxiway] 2; cleared the ZZZZZ3 for takeoff Runway XYL; then cleared the ZZZZZ2 off of Runway XXR at [Taxiway] 1. I shipped the ZZZZZ1 to Departure; and once the ZZZZZ3 was airborne passing behind (diverging from) the ZZZZZ1 shipped them to Departure. Last; I shipped the ZZZZZ2 to Departure. As I migrated through the rest of my work; I noticed the ZZZZZ2 was turning northbound; and exclaimed something along the lines of 'Where are they going' while reaching out to them.They were still with me. I quickly; while they were climbing to meet the MVA explained that they were to have gone straight out via ZZZ radial and said turn left immediately heading 310; traffic 11 o'clock and 2 miles opposite direction VFR traffic at 500 feet below (now 1;000 ft. above the VFR); and said continue left turn heading 260; maintain 3;000 ft. I then advised possible pilot deviation and to call the Tower; Approach Control would provide the phone number. I did not mention the ZZZZZ3 traffic; as the VFR was closer and I was providing Tower visual separation between the ZZZZZ3 and the ZZZZZ2. I also did not state low altitude alert as the aircraft; once acknowledged on frequency was climbing through it. Once they were turned to 260 I told them once again to contact Departure. I am not sure why the aircraft turned when the SID depicts straight out to ZZZZZ4. I guess an RNAV SID where I could say 'RNAV ZZZZZ4' would solve this one.