If the ancient one awakens after an investigator dies in the mythos phase, does the player draw a new investigator?
The player draws a new investigator and continues play.
From the FAQ (emphasis mine):

If an investigator is defeated, the player controlling that investigator chooses a new investigator at the end of the Mythos Phase.
...
After the Ancient One has awakened, when an investigator is defeated or devoured, the player controlling that investigator is eliminated.

Resource guide says the same thing.
This isn't MtG, there isn't a stack.  My understanding of the &quot;wait til the end of the mythos phase to start new guy&quot; rule is to prevent any/all confusion about what actions (and interactions) that person can take.