Trying to build a white noise microphone blocker

Truly, the cheapest way depends on what the OP has on hand to work with. The solid state generators do have some "cool" factor. I've even seen a circuit that implements "fractal chaos".
http://www.scholarpedia.org/article/Transistor-based_chaotic_oscillator

Here, the FM stations are jam packed from one end of the band to another, so it wouldn't work where I am, either...