A microcontroller like a PIC and Atmel. You would need a program and programmer, need to wire it all up. But would be simple.
One input for the quiz master (reset the light and start the timer), 6 inputs for the contestants, 6 outputs for the lights (or LEDs), one output for the buzzer.
You would need LEDs, momentary switches and a buzzer. How big an area is this (How loud does the buzzer need to be)?
In the software when you first turn it on, have it watch the quiz master switch to pulse the timer increments (say 5 seconds each click) to bump the delay timer (3 clicks will be 15 seconds).
Maybe I will see if my son wants to make one? If so, I will share it with you.