By the way, don't underestimate the power of analog, we might call it old-school, but the only old part about it is they're primordial laws of physics at work!
Seems doable for sure, there may be a use for a small drawer full of cmos/ttl chips after all.
Studying the aforementioned ttl book, a random pattern could be accomplished with say a 12 x 16 array (192 leds). In a common anode configuration, a random pattern could be configured by sinking the current thru a 1 of16 bit decoder chips (74154). That would give you 12 blinking leds at a time, or you could cascade two of them for 6 leds.
All the inputs to the decoder chips could be handled by six 8 bit shift registers ('164's or '595's?), which in turn are driven by a serial randomize'r circuit. The serial randomize'r is an 8 bit shift register combined with an xor gate for 15 different states (if grouped for 16 leds).
Anonophiles's cicuit could be possibly be used for driving the cathodes? How would one do the clocking of the randomize'r, and latching/synchchornizing of of the cathode's