At its core, this is a question about Self Reference. I want to start by reducing this to propositional or sentential self-reference, rather than dealing specifically with questions and answers, then I want to point you in the direction of how you might usefully start to address this within a framework of implicit hierarchies of sets or predicates.
Curiously enough, I've suggested an analysis of the propositional form of asking questions on this site in the past. The idea is that a well-formed question is a particular act querying a proposition that may or may not be true - we divide the scope of possible assertions into those that imply the truth of some proposition and those that imply its falsehood, and we are desiring someone to present us with a proposition in one of those categories that we say suffices to have answered it.
The paradigmatic "unanswerable" question seems to be the liar interrogation:
"Is your answer to this question false?"
It would seem impossible to give a consistent answer to this question. To answer in the affirmative is to assert that one's answer is false, thereby satisfying the requirements needed to make the underlying proposition true, but this runs contrary to what we said (similarly with the negative).
But this is not particularly difficult to schematically break down into a propositional form. One could use a different form of indexical reference here to break down exactly what "your answer to this question" is - for example, "the statement you make within some space of time subsequent to my having made this utterance made in the intention of providing me with an answer given the intention you take me to have had in asking it".
So the "Question" part of this is a meandering but ultimately tractable way of hooking into a deeper phenomenon, which is propositional or sentential self-reference. Really, all I'm doing in asking the Liar Interrogation is prompting you to state a conventional liar sentence (or its negation, which is in fact the same proposition), and that is a well known topic of interest in philosophical analysis.
Now, let us try to formulate the "meta" question - can one non-arbitrarily assign a sequence of priorities to formulations of a self-referential proposition?
And, as has been well demonstrated, yes, you can. The key here is to understand the application of a Fixed Point theorem (relevant SEP section) for an evaluation operation. If, for example, you want to try to cash out this idea of reference in terms of Truth conditions, and by introducing a predicate for affirming the truth of sentences into our language, you can construct incremental refinements of languages including Truth, and respecting our definitions for what it means for a given sentence to be true (e.g.
Tr('x') iff x).
These refinements result in a hierarchy of languages, each of which has a larger body of admissable statements, which we can index using an ordinal number. And, eventually, by considering these sets of sentences and the operation of incremental augmentations of truths, we reach a Fixed point, where our operation of adding new truths has stabilized and we're not adding anything new, such that
Tr('x') = x, and this is a language exhibiting self-referential properties.
The existence of such fixed points falls out as a mathematical theorem from how we've gone about incrementally refining our language. What's particularly interesting is that as part of this proof, we don't only show that fixed points exist, but we also prove that there is a least such fixed point. That is, sentences of a relevant self-referential nature can be connected ordinally to a sequence of operations of incremental refinements of language, and it is possible to identify the first ordinal level at which our required self-referential behaviour occurs.
Our process is ordinally indexed from the get go, so all we need to do to rank our "answers" in terms of this ordinal sequence, et voila. Non-trivial, non-arbitrary orderings of self-referential statements in terms of the languages that sufficiently interpret them.
I've been a bit fuzzy about this because I don't really intend to go through the nuts and bolts of exactly what kind of self-referential behaviour you're looking to exploring further. However, I would be confident that the most correct answer to your question is "no, it is not necessarily arbitrary, and indeed it is quite important to understand why there is an appropriate ordering involved in current theories of propositional self-reference".