How about this?
In the next movie we find out that Harrison wasn't really Khan but another Augie Doggie who used Khan's name. The real Khan is stored away somewhere like the Ark at the end of Raiders.
Instant Retcon's gonna get you ...
I was going to do a separate post to suggest just that.
Maybe Khan had the name switched so an enemy would kill the wrong person, or Harrison claimed to be Khan, so he could be more respected/valued, and made up some lie about it waking up.
Or perhaps son of Khan Noonian Singh?
Keith Hamilton Cobb
of Andromeda fame would have been an excellent Khan