GA student pilot reported a runway excursion just prior rotation on takeoff. Pilot regained directional control on adjacent taxiway and was cleared to taxi to hangar where a mechanic noted cosmetic damage to left gear. No determination of why the aircraft suddenly veered left.

Narrative

Student pilot practiced takeoffs and landings dual instruction the morning of Date. After practicing traditional landings and go-arounds; [the] instructor was dropped off at [the] hangar for first solo flight. Student executed first solo take off; trip around airport pattern and full-stop landing in a successful and skilled fashion. Taxi back to Runway XX for second solo. After clearance; full throttle applied; right rudder as needed; all gauges and systems indicating appropriately. Just prior to the intersection with Taxiway XX; plane veers to the left just prior to full rotation speed; and rumbling sensation was described by student pilot on left side of plane just aft of the pilot. Student pilot successfully pulled throttle idle while trying to maintain control of the aircraft. Aircraft exited Runway XX through the grass median; and onto Taxiway XY. The plane briefly left Taxiway XY to the grass before the pilot was able to regain full control and return the plane to the Taxiway XY in a south facing direction. Plane came to rest on Taxiway XY while still in motion and pilot regaining control. Discussion with ZZZ tower followed; and pilot verified plane systems [were] able to return aircraft to hangar. The plane was returned to the hangar by the student pilot; normal shutdown via the checklist; and inspection with the instructor.After a successful and well executed first solo; the incident occurred on the second solo. Student pilot described the situation above. Factors contributing are unknown; but [the] plane was examined after the incident. The question being what caused the plane to abruptly veer left after an initial centerline beginning. ZZZ safety surveyed [the] field; for potential causes; as well as identified the damage to one runway light. Potential causes include mechanical dysfunction; field obstruction; or error in pilot control input. Final cause is unable to be determined; but actions include: Plane was inspected by [a] mechanic; and cosmetic damage was observed to left gear strut probably from impact of runway light with yellow paint noted.Student pilot has a clear training history; currency; and practice prior to the solo today. No prior difficulty with takeoffs or landings. Student pilot reviewed potential input errors with CFI; and will continue to be vigilant in control habits as he has in the past.Student pilot called and discussed incident with tower ATC; and provided any information needed; as well as contact information. Plan is to continue satisfactory completion of PPL with CFI.