Below is the Mind Flayer's Mind Blast feature:
Each creature in that area must succeed on a DC 15 Intelligence saving throw or take 22 (4d8 + 4) psychic damage and be stunned for 1 minute. A creature can repeat the saving throw at the end of each of its turns, ending the effect on itself on a success. (Monster Manual, pg 222)
Suppose a character was knocked unconscious by an illithid's mind blast, and then got healed less than a minute later. What is the correct way of handling whether or not they're stunned?
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The stunned condition ends when they're knocked unconscious, and reviving them means they're not stunned.
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The stunned condition continues, in addition to rolling death saves they should be rolling Int saves to shake off the stun.
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The stunned condition continues, but an unconscious character can't make an Int save, so they should roll an Int save upon being revived to see if they shake it off.
Best Answer
The Stunned condition lasts until a minute passes (or if you succeed on the saving throw); in general, conditions tell you when/how they end
The section on "Conditions" states:
In this particular case the Mind Flayer's statblock states:
Unless something explicitly says otherwise, the stunned condition that Mind Blast causes will not end until the 1 minute is up. As user @Ryan Thompson points out, there is further evidence that conditions persist even through the more extreme case of death. The PHB errata states:
There would be no reason to reduce a creature's exhaustion level upon resurrection unless death did not reduce it to 0 already. Thus exhaustion levels persist even through death, let alone just unconsciousness, this would similarly apply to other conditions as well.
It is noteworthy that while unconscious you would continue to make saving throws to end the stun from Mind Blast. This could end the effect before the 1 minute is up, and is possible because ending the Stunned condition requires an Intelligence saving throw -- both the Unconscious and Stunned conditions do not make you automatically fail these.
If you wish to remove the stunned condition I refer you to the question "How can I end the Stunned Condition?"