When people talk about "fundamental" or "inherent" randomness in the context of quantum mechanics, the technical meaning behind this is Bell's theorem, which tells us that there are no local hidden variable theories explaining the results of quantum mechanics.
A "local hidden variable" theory is basically the classical idea of how the world works - everything has a list of well-defined properties, like position or momentum, and there is a "true" precise value for each of these at each time, and the laws of physics in principle determine the precise value at each other time from those at one instant. "Randomness" in this classical world is incidental, arising from incomplete knowledge, imperfect measurement devices, etc. When you flip a classical coin in the exact same way, it will always yield the same result. The "randomness" is just because humans are extremely bad at the level of consistency required to flip it "in the same way" again. The belief that there is a definite value for each property at all times is also called realism.
Bell's theorem says that quantum mechanics is incompatible with local hidden variable theories. No such theory can ever predict the results that we do, in fact, observe. (Hunting for and closing "loopholes" in our experiments that might make it possible to argue we don't actually observe the violations of the Bell inequalities that rule out local hidden variable theories is a somewhat active niche I won't get into here.)
So "fundamental randomness" really is supposed to mean "no hidden variables" - before you measured a particle's momentum, it didn't have a definite one. The quantum state is not a list of numbers with definite values for properties we can measure, it is merely a list of probabilities. To say this is "fundamental" is to say that it is impossible to explain these probabilities as just arising from our lack of knowledge of some underlying definite variables, i.e. it is the content of Bell's theorem. The claim is that the uncertainties and probabilities of quantum mechanics are really features of the world, not features of our inability to comprehend it.
For completeness, let me mention that Bell's theorem gives you a way to preserve belief in hidden variables - instead of abandoning realism you can choose to give up locality, roughly speaking the notion that things cannot instantaneously affect the state of other things separated from them in space. This is what Bohmian mechanics does, but it is far from being the dominant viewpoint among physicists. Although there is a plethora of different quantum interpretations, which are effectively ontological frameworks trying to explain how to think about a world that is not classical and mechanistic, most of them choose locality and abandon realism - which is why you'll often hear that "quantum mechanics says the world is fundamentally random".