Repeatability -- the same sequence should be produced with the same initial values (or seeds). This is vital for debugging etc.
Randomness -- should produce independent uniformly distributed random variables that pass all statistical tests for randomness.
Long period -- a pseudo-random number sequence uses finite precision arithmetic, so the sequence must repeat itself with a finite period. This should be much longer than the amount of random numbers needed for the simulation.
Insensitive to seeds -- period and randomness properties should not depend on the initial seeds.