Is there a standard definition of Quantum Randomness?
The randomness used in quantum computing comes from quantum mechanics and is postulated to be an inherent 'randomness' of nature (and one of the problems Einstein, for instance, had with QM). If you take a state like

$|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$ 

and measure it in the computational basis, there is absolutely nothing you can do to predict the result (assuming our standard pictures of QM is not horribly wrong). Half the time you will measure $0$ and half the time you will measure $1$. Of course, if you measure in the Hadamard ($|+\rangle$, $|-\rangle$) basis, then your result will always be $+$.

I don't think there is a standard notion of quantum randomness, mostly because how random a state looks depends on your choice of measurement. Once you choose a particular basis to measure in, then you might as well start talking about the classical probabilities associated with the results and then you can do your standard classical tests of randomness.

Usually when someone talks about the 'quantumness' of a state, they try to capture ideas like entanglement.