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Yep, a really important point here. Randomness, sensu strictu, is not a property of existing things but rather a characteristic of the process by which existing things come to be. A random process can, given enough time, generate any existing thing, or anything that can be imagined. And most existing things/imagined things (Chaitin's omega being an exception) can equally be generated by either a random or a deterministic process. Its precisely for this reason that it may make sense to use as a criterion of randomness the breadth of a process's ability to explore possibilities rather than any particular one of its products.

There is, though, a more relaxed/common use of "randomness": the absence of any discernable order (in either space of time) or, more generally, the absence of any ability to make any useful predication about a particular element of any array (in either space and time) given knowledge of the rest of the elements. A sequence of numbers produced by a random number generator has this property (despite being generated deterministically). A completed Sudoku puzzle doesn't because the properties of each element (the number at a particular location) depends on the properties of other elements. To put it differently, any correlation among elements in an existing/imagined thing makes it non-random in the more common/relaxed sense of randomness. And, in this circumstance, there can be varying degrees of randomness (more or less correlation).

The upshot: "randomness" is, like most human ideas, an "onion," with nested different meanings that need to be peeled apart. A random process can generate an uncorrelated product that will appear random to an observer. But one can have uncorrelated products that result from non-random processes, and one can have correlate products that appear random to an observer. And one can have correlated products that result from random processes that may or may not appear random to an observer.