Meaning uncertain. According to a Georgian source, the name is of Greek origin and means something along the lines of "according to the moon" or "spoken by the moon". The closest Greek name with that kind of meaning would be
Menagoras, which is properly transcribed as
Minagoras when you follow the transcription rules for modern Greek.
However, that name was quite rare in both Classical Antiquity and Late Antiquity. It seems unlikely that such a rare name would somehow find its way into Georgia and subsequently manage to survive there into modern times. On top of that, the Georgian form of the name should actually be either მენაგორა
(Menagora) or მინაგორა
(Minagora). After all, the Georgian form of the similar Greek names
Anaxagoras and
Pythagoras is ანაქსაგორა
(Anaksagora) and პითაგორა
(Pitagora) respectively.
All in all, I would say that until we have more detailed information about the etymology of Minago, we should be very cautious about accepting the Greek etymology provided by that one Georgian source.