According to the airline transport pilot, during the takeoff roll the airplane briefly exited the runway to the left after the passenger inadvertently pushed the left rudder pedal. While correcting for the runway excursion, the right main landing gear struck a taxiway sign, as the pilot controlled the airplane back on to the runway. The pilot and passenger reported that they did not observe any damage from their cockpit seats, and continued the takeoff roll and departure. The pilot reported that after reaching their destination, during the touchdown roll he noticed the right main landing gear tire deflated and affirmed that "the right wheel dug in and threw up debris that damaged the right elevator." However, photographs taken at the departure airport depict a taxiway sign with blue paint and metallic particle transfer on the sign which appears to be the same blue color as the accident airplane. Further the photographs show damage to the sign consistent with the damage to the airplane's elevator. There were impact impressions on the taxiway sign's black metal frame that were consistent with the damage sustained to the right side of the airplane's elevator. The pilot reported that he used duct tape to make repairs to the airplane, took off and landed at their final destination airport without further incident. The airplane sustained substantial damage to the elevator. The pilot reported that there were no mechanical failures or anomalies with the airplane prior to departure or during the flight that would have prevented normal flight operation.