How to build a noise generator based entirely on logic gates?
How about one of these: -



Taken from here.

It is described as a white noise source: -


  24 Stage Linear Feedback Shift Register (LFSR)
  
  This circuit can be used to generate a pseudo-random sequence of 0's
  and 1's -- that is "white noise" or "static." The schematic shows a 24
  stage shift register with XNOR taps at registers 7, 16, 22, and 24
  using the Fibonacci configuration. With these taps, the circuit
  generates a "maximal length sequence" of pseudo random values which
  repeats only after (2\$^{24}\$ - 1) clock pulses.


Or maybe just use an analogue one based on the emitter-base breakdown voltage of transistor Q1: -