I have been asked to create a circuit that drives a set of LED's that when switched off uses a random timer from ~1 - 3 seconds before the lights go off. This will be used in a light signal box for a motorcycle roadracing start signal. The starter will turn the box on (3 rows of LED trailer tail lights for example) then when the bikes are all grided up and ready, the starter will flip the switch to "off" and then a random time between ~1 - 3 seconds the lights go off and the bikes launch. Can you all point me in the right direction? Thanks!
A small microcontroller is the most appropriate choice for this. Yes, it's possible to create a random analog circuit, or latch a high-frequency counter to create a random digital value, but it's going to be more fiddly and many more components than you want to deal with.
Any MCU will happily do this. I'm guessing you are not very familiar with microcontrollers, and therefore I would recommend an Arduino due to the large volume of community support and existing resources.
If you really want to do this with discrete logic, use a ring oscillator (high frequency) to drive a small counter. When the button is pressed, latch the value, load it into a downcounter, and start the downcounter. When the downcounter reaches zero, the lights go off.