Tower Controller reported a student pilot landed off runway after an engine failure due to fuel starvation while in the traffic pattern.

Narrative

I had been working Aircraft X; a student pilot with an instructor on board; in the left closed pattern on Runway XX. I had two jets taxiing out and had just finished coordinating a release for one jet to ZZZ1. Shortly before; I completed a scan of the runway prior to Aircraft X touching down on Runway XX for their option clearance. Shortly after acknowledging a request from a jet departure taxiing out; I heard Aircraft X call out on local frequency that they had just experienced an engine failure or engine power off and were making a landing. I acknowledged the transmission and immediately returned my attention to the runway. It took me a moment to find Aircraft X as they were in an unexpected position due to their engine failure. I located Aircraft X in the grass off the departure end of the runway just as they called me again to tell me they were in the grass and had no injuries. I asked if Aircraft X would still like a response from ARFF (Airport Rescue and Firefighting); they replied that they did. I activated the crash alarm at the local position; and the vehicles came out shortly after. The airport closed Runway XX in order to serve the disabled aircraft; and I was able to move my jet departures to Runway XY. Not long after; Aircraft X was able to taxi back to the service ramp under their own power; and the airport was able to re-open Runway XX after completing a FOD check. We later received the necessary information from the pilot; stating they had forgotten to switch fuel tanks and the engine had failed from a lack of fuel while in the pattern. After listening to the tapes; I felt it was an error to ask Aircraft X if they wanted me to start the ARFF response; I should have done that without asking. I was a little caught off guard by the whole incident and did what I thought was right in the moment; but I would do that differently. When the ARFF trucks came out; they seemed to be confused about which end of Runway XX the aircraft was on. I may have contributed that confusion by stating the aircraft was; 'off the departure end of Runway XX by the approach lights.' If I would have said; 'by the YY approach lights;' that might have been more clear.