There are two circuits that come to mind for this. The first is a simple flasher circuit that runs for more than a year from a simple AA cell.
http://www.discovercircuits.com/DJ-Circuits/1vled3.htm
There is another flasher on Dave Johnson's site :
http://www.discovercircuits.com/H-Corner/led-flasher.htm
I built this latter one and found it worked very well. I think, though, that it can't work with LEDs needing more than about 2.5 volts, which would tend to exclude white leds. The battery lasted a very very long time.
The other circuit is the simple pseudo random sequencer such as on page 16 of this link:
http://www.cs.ucsd.edu/classes/fa05/cse140L/Lecture_1026.pdf
or in section 5 of this:
http://people.clarkson.edu/~ortmeyer/ee211/sequential%20logic/Fundamentals%20of%20Digital%20Electronics.pdf
One approach is to wire up several of those LED flashers with very different time periods, all quite lengthy but all independent of each other, and simply have each one feed a separate fiber. Would the result look random? perhaps.
Otherwise, another approach would be to have the PRN sequence generator drive an analog 1 to 16 Demux, with each output feeding a separate fiber LED. Perhaps each output would drive one of those LED flashers.
Of course, others will advise, rightly so, that the simplest hardware would have a PIC or AVR with quite a few outputs, each driving an LED. But then someone has to write a program to toggle those outputs randomly, which is not hard if you are PIC programmer.