I like circuits like these that do things randomly.
The best and simplest way to produce a truly random output (that I know of) is through the use of relaxation oscillators. It uses discharging capacitors that discharge when gas in neon bulbs ionize - which is random.
Neon bulbs that flicker (due to lower pressure of neon gas inside them through their long use) probably would be ideal. You can see this effect on those little battery operated neon flickering candles you can get at hobby stores. You can also sometimes see this flickering effect inside the orange glowing rocker switches on AC power strips from the neon bulbs inside them - after several years of use.
Take a look at the Nothing Box by Mike Ellis:
Nothing Box
For this application, neon bulbs would be better to use but if you still want to go with using LEDs, you would have to connect them across each group of resistor, capacitor and neon bulb group as illustrated in the above circuit and just cover the neon bulbs to cover their lighting. You just have to be sure to limit the voltage and current discharges through each LED so that you don't burn them out. Use a current limiting resistor for each LED. I would start with an experimenter board first and hook up one LED, neon bulb, capacitor and resistor with a 12V source first. I'm not exactly sure on the configuration of hooking up an LED with the relaxation oscillators, but I would experiment first.
Here's another circuit, but not truly random outputs, it just appears that way since it's driven by the 555 timer:
8 Random Flashing Leds Circuit
Google these for more complex ways:
random number generator ic (like how a scientific calculator generates random numbers for statistics.)
Random Number Generator by IC 4017 and 4011 | Circuit Project Electronic
random numbers via quantum mechanical randomness [diode noise]
Hope this helps.
Jeff