I've never heard this definition before.
Doxa = opinion, belief
n
1. a seemingly absurd or self-contradictory statement that is or may be true: religious truths are often expressed in paradox.
2. (Logic) a self-contradictory proposition, such as I always tell lies
3. a person or thing exhibiting apparently contradictory characteristics
4. an opinion that conflicts with common belief
[C16: from Late Latin paradoxum, from Greek paradoxos opposed to existing notions, from para-1 + doxa opinion]
Collins English Dictionary – Complete and Unabridged © HarperCollins Publishers 1991, 1994, 1998, 2000, 2003
A paradox is usually understood to be a logical contradiction. Of course what is believed to be logically contradictory may well be up for debate (this may be what you meant - i.e. dependent on opinion).
For a contradiction to be a paradox it needs to look true to most people. Most people can see that "A and non-A" is an obvious contradiction and therefore necessarily false so it's not a paradox.
The argument that Achilles will never catch up with the tortous is a paradox only because we all believe that he will catch up. Yet, it's not enough to say that. The crucial point is that most people cannot identify the error in the argument. They think it's wrong, it's in their face, and yet they can't explain how it's wrong. So here, even if the argument is wrong, it is still justified to call it a paradox.
In any event Togo's godel-box thought experiment produces a counterintuitive result (what he calls implausible) but there's nothing logically contradictory about it.
It depends what you include in the argument.
Proposition A - The universe is deterministic therefore the Gödel Box correctly predicts decisions.
Proposition B - It is obvious that a subject informed in advance of the Gödel Box prediction of the action he is to decide on will be able to change his decision, thereby making the prediction false.
A would be the opinion (of hardcore materialists) and B the paradox going against A.
For all, B is not a paradox if it's not their opinion that A is true.
However, since most hardcore materialists believe determinism is true they may also take A to be true. B seems to go against A. Is there a contradiction between A and B? Maybe yes, maybe no but they do look contradictory. So B on it's own will be paradoxical to hardcore materialists only to the extent that their opinion is that A is true and they can't explain exactly how B is wrong (if they do then the paradox is solved and no longer a paradox for them).
Here the Gödel box thought experiment goes indeed against the opinion of the hard-core materialists here,
I guess I'm a 'hard core materialist' but nothing about the Godel box TE goes "against" my 'opinions'.
No because it's your opinion that a Gödel box could predict opinions. So the paradox for you to explain should be proposition B suggesting that A cannot in fact be true.
EB