Probably the idea that the Enterprise crew came together in a time of crisis. And Uhura having a much more prominent role than in TOS.
I'd mention the idea that the crew attended the academy together, but that almost came to light in the early 90's with Star Trek: The Academy Years
and isn't an invention of the '09 reboot.
EDIT: And probably the idea that Khan is Star Trek's version of The Joker or Moriarty, an arch nemesis for Kirk and crew.