I'm not sure of the precise name of this box, a friend who is a professional mechanic called it a 'brake light relay'. I believe this is also responsible for reporting a bad brake lamp. At first when my 3rd brake light was going out, the lamp check light on the dash would light up (it would go on and off depending if I hit a bump). After time, the 3rd brake light was out completely and the lamp check light on the dash would NOT light up. So it seems to be responsible for both providing current to the brake lights and reporting a bad brake light.