"Randomness" versus "uncertainty" Highly rated PhysicsSE contributor @CuriousOne regularly makes the following claim about quantum mechanics (e.g. here):

There is no randomness in quantum mechanics, there is only uncertainty. 

I want to know what this is supposed to mean.
 A: There is no randomness in quantum mechanics, there is only uncertainty. , as stated, whoever may have said it.
Mathematical definition of randomness:

The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events.

So by this definition, mathematically, randomness is defined wherever probability distributions can be assigned to expected outcomes.
As quantum mechanics is par excellence a probabilistic theory, i.e. probability distributions are assigned to measurable variables from solutions of relevant differential equations, this mathematical definition of randomness is not the one used in the statement. It must be the everyday concept in the beginning of the link:

Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination.

So the statement There is no randomness  can be applied to any event that can be predicted/estimated from a probability distribution, and not only in  quantum mechanics. When a mathematical probability distribution exists, there is only uncertainty for a specific measurement , given in terms of the probability for that outcome.
So the statement should be :
There is no randomness where a mathematical probability distribution exists, there is only uncertainty .
So the original is a statement valid in a discussion where deterministic classical theories which predict a single measurement value as the solutions of the appropriate equations are being discussed, whereas quantum mechanics ( or any probabilistic theory)  predicts an uncertain value with a given probability. 
The above exposition goes to prove that the context where a  statement is  made is as important as boundary values for  physics solutions.
A: The core of your question is subtle, so I'll be careful in how I set up my answer. 
In my understanding of quantum mechanics, wave function collapse is the closest a physical process can be to the mathematical idealization of a random variable. However, before the collapse, a complicated many-body process, the wave function evolution of the system is deterministic (unitary). One could say that uncertainty is what characterizes the wave function before measurement (measurement being viewed as a kind of dissipative process), while "randomness" characterizes the probabilities of making various measurements, implicitly after any decoherence necessary for measurement has occured. 
[Addendum: In the context of quantum computing, it's important to find a set of measurements (quantum numbers used to read the output of the computation) that align as much as possible with the set of (meaningful) final states that the algorithm could produce. If there is some misalignment between the measurement axis and these final states, noise is generated when reading the output.]
