The save has already been attempted
You ask two questions and make a fundamental misunderstanding.
The situation that you describe is that the PC has been hit with a mind flayer's tentacle attack and has gone unconscious from the resulting damage. They are later healed. You ask (2) whether they are still grappled, and thus (1) would they have to attempt the save against stun when they regain consciousness.
Yes, they are still grappled, as Eddymage explains. However, the PC should have already attempted the save against stun before they regained consciousness. That save was attempted on the mind flayer's turn as a consequence of the attack.
You might be assuming that if the PC has been made unconscious, they do not attempt the save against Stun. But nothing in the unconscious condition prevents being stunned.
Or, as V2Blast suggested in a comment, you might think that being Stunned means an automatic failure on Intelligence saving throws (as it actually does for both Strength and Dexterity saves). However, nothing about being Stunned interacts with the Int save (in particular, nothing about it would either guarantee success or failure on the save).
Thus the PC has to attempt the save on the same turn they are rendered unconscious. When they are later healed and regain consciousness they are still grappled, but they have already either failed or made the Int save and thus are already either stunned or not. If they were stunned, they will remain stunned until the grapple ends (or until they make the save, or a minute has passed), regardless of whether or not they are conscious.