Lazav, Dimir Mastermind can become a copy of
Jace, Vryn's Prodigy.
There is no technical step-by-step process for changing characteristics during a zone change–there is simply one simultaneous action where an object that has a certain set of characteristics leaves the old zone, and a new object with a different set of characteristics enters the new zone.
So
Jace, Telepath Unbound being put into the graveyard would not trigger an ability that triggers “whenever a creature dies”, because that trigger condition is looking to see an object that is a creature on the battlefield leaving that zone for the graveyard, and
Jace, Telepath Unbound being put into the graveyard will not match that description.
Lazav, Dimir Mastermind, on the other hand, triggers “whenever a creature card is put into an opponent's graveyard from anywhere”, meaning it's looking to see an object that is a creature card arriving in the opponent's graveyard from anywhere, and
Jace, Telepath Unbound being put into the graveyard will match that condition.