Air traffic controller reported working a departing aircraft that loaded the wrong data in their FMS resulting in flight towards terrain. ATC issued an alert and turned the aircraft away from the terrain.

Narrative

Aircraft X departed ZZZ RWY XX around XA:15Z on the ZZZZZ SID. Aircraft checked in; was radar identified; given a climb to 6;000 and a query for a PIREP due to low cloud layers. The pilot responded with a tops estimate and the new altitude. After the tops estimate was passed to me; an aircraft departed ZZZ1 and I focused my attention on that aircraft. I noticed Aircraft X making a turn to the north; I issued a vector to a 270 heading and issued the MVAs in the area. I noticed the aircraft was turning far tighter than normal and that is when the pilot stated they had loaded the wrong runway and were turning on course for the RNAV fixes for RWY XY.The pilot had departed XX and turned to the fixes for the wrong runway (XY); putting the aircraft into higher MVAs while there was also a low cloud layer.Pilots need to start stating at some point; either to GND (Ground); LCL (Local Control); or the first departure controller their runway assignment and RNAV fix navigating to on an RNAV SID.These events have happened multiple times and can lead to a very dangerous situation with traffic; and/or terrain which could lead to a crash.