The term is called Argument from silence. It is when a person uses another person's silence (or absense of a statement) as an information in itself and not as just silence.
According to rationalwiki.org:
An argument from silence is an informal fallacy that occurs when someone interprets someone's or something's silence as anything other than silence, typically claiming that the silence was in fact communicating agreement or disagreement.
It is considered an informal fallacy because it is not a solid argument that support its conclusion with actual proofs. The silence could be due to various other reasons, thus the conclusion might as well be wrong as it might be correct. Take the three hats puzzle as an example, the prisoner's interpretation of the silent prisoner's silence could be wrong, the silence might be because:
- The prisoner is mute.
- The prisoner is blind, so he can't possibly deduce his hat color and give an answer even if he could.
- The prisoner might be slow-witted, so it should take him a longer time to deduce his hat color, thus the delay/silence is unreliable.
In all the above situations, the silence does not definitly indicate that the silent prisoner couldn't identify his hat color because of the arrangment of the hats alone.
But in the context of puzzles, one usually uses the best case scenario, where the argument from silence becomes a solid argument (the silence will only mean one thing, thus we use that thing in further deductions).
Note: The term, however, for @Bass comment is Arguments from ignorance. They're basically the same thing, except that AFS is interpreting silence, whereas AFI is interpreting ignorance of a matter/subject.