When considering random generators and random number generators for security purposes you have to be extremely careful. To answer your question, there are many tests that you can carry out on RNGs, for example see the links offered here: http://csrc.nist.gov/groups/ST/toolkit/rng/batteries_stats_test.html
Please note that I am assuming that you are actually asking about "how to test random number generators for password generators" as opposed to the question "how can I measure the randomness (or entropy) of a given random array of bits" - which has been answered above.
Note that a random number generated can be easily used to generate passwords: generate several bytes of data and transform this data into the password using some transformation of your choice.
I wish to draw your attention to the notion of cryptographically secure RNGs. While the Mersenne Twister is generally considered as one of the better RNGs you can use because it is efficient and has very large periods before repetition, etc..., it is NOT considered to be cryptographically secure.
A cryptographically secure RNG algorithm is important for security sensitive applications (such as password generators). Without this property, an attacker can infer the current state of your RNG and be able to predict your next password.
Now, RNG algorithms are all pseudorandom (unless you use a truly random source such as http://www.random.org or a Quantum Random Number generator card). Given knowledge about the current state of the algorithm, we can predict perfectly the next "random" number. An attacker that infers this state would therefore be able to predict your next "random" password.
Besides the myriad of tests for a good Pseudo-Random Number Generator, also make sure that you check that it actually is cryptographically secure. Otherwise go for a true random number generator source.