Last year I wrote a chapter for O’Reilly’s book Beautiful Testing (ISBN 0596159811). The publisher gave each of us permission to post our chapters online, and so here is Chapter 10: How to test a random number generator.
Update: The chapter linked to above describes how to test transformations of a trusted uniform random number generator. For example, maybe your programming language provides a way to generate numbers between 0 and 1 uniformly, and you have written code to transform that into a normal distribution. That’s the most common case. Few people write their own core RNG; most bootstrap the core RNG into other kinds of RNG.
If you need to test a uniform random number generator, I can help with that.