I want to try and give an answer to this question from a "objective Bayesian" perspective. Objective Bayesianism is the idea that probability measures should be thought of as corresponding to states of knowledge that an observer might have. The "objective" part refers to the idea that in some situations there is a "correct" state of belief that one should have, and therefore a correct prior to use.
For an objective Bayesian, to say something is uniformly distributed on [0,1) means that they don't know its numerical value (beyond it being at least 0 and less than 1). The probabilities do not represent randomness in this case, but just how likely the person thinks the value is to fall into a given range. This can be formalised in terms of the odds they would accept on a bet, or in several other ways.
So far there isn't any notion of "random choice", or randomness at all - the person could simply be reasoning about an unknown fixed value. But suppose we want to construct something, physically, such that our state of knowledge about it is the uniform distribution. That means we need to create a value, somehow, such that (i) we don't immediately know it, and (ii) according to the knowledge we do have, it is as likely to be within one sub-interval of [0,1) as another of the same length.
It's kind of hard to think about how you'd do that for a real number (and we can get into all sorts of gnarly issues about how accurately it's possible to measure something), so let's switch to a more familiar example: we want to choose a number that's uniformly distributed on the set {1,2,3,4,5,6}.
That means we want to create something such that (i) it has a definite integer value between 1 and 6, (ii) we don't immediately know what that value is, and (iii) any value between 1 and 6 is as likely as any other.
The obvious solution to this is to roll a die. Assume for now that we roll it behind a screen, so we don't know what value it has. Then we have a value between 1 and 6 that we don't know, and if we assume the die is properly made and rolled well, then we don't have any reason to think any value is more likely than any other.
We don't have to roll the die behind a screen either - we can just say that before we roll the die there is still a fact of the matter about what face will come up, it's just that we don't know what that is until we see it. After all, when you roll a die you're just imparting some momentum to it. The face that comes up depends on that initial momentum in a very complicated way, depending on collisions with the table and with air molecules and so on, but in principle it could be calculated - it's just very hard to do so.
(One can argue about whether that's true or not - are quantum effects important in the rolling of a die, and if so are they "truly random" in a way that classical mechanics isn't? What about the free will of the person rolling the die? - but I will neglect these arguments because they're not really crucial to the point. If you prefer, instead of a die you can think of a random number generator on a computer, where none of those arguments apply.)
For a real number between [0,1) you might do something like spin a disk with the numbers from 0 to 0.9999 marked on the circumference. Or you could roll a 10-sided die a large number of times to get the decimal expansion, e.g. 0.1853724... (Both of these have the issue that you can't really get an infinite number of decimal places, although the second one allows you to get as many as you need for any given purpose.)
In short, I'm proposing that "a single random choice" could be defined as something like "a process designed to produce a result that is unknown for a given observer, and such that for that observer their subjective probability distribution has a given form."
