I've had this happen to two different cars before. There's a part specific name for it, but it's nothing more that a rubber grommet/pad that seperates the activator behind the brake pedal pad to the arm of the pedal. When the two connect when bake is applied it completes a circuit connection which turns your brake light on. When the pressure is disengaged the circuit is released so the light turns off. Since it's a little piece or rubber it's prone over time to dry up and break off (I bet if you looked on the driver's floor board you would have found a piece of rubber). Just any little piece from your 'junk drawer' or kiddie's playbox can be used so long as it's small enough to fit where it goes. Good luck!