Circuit that randomizes shut off time to LED's from 1 - 3 seconds when switch is thrown
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.