Other mistake: In the scene after Jake first links to his Avatar body and they are getting ready to put the Avatar's to rest, Grace gets the human scientists to 'scat' and shuts two large gates behind them. There is a CGI error with Grace's left hand. Her hand is not holding the gate handle but is pulling it shut anyway.
Other mistake: During the battle between the marines and the Navi, colonel Quaritch shoots down Trudy's ship. Despite her cockpit having bullet holes in it, somehow she is still able to breathe. Unlike Quaritch when Jake sticks the knife into his AMP.
Other mistake: When Michelle Rodriguez is flying the copter she doesn't have a mask on when they are at the tree of life to shoot down the tree. The machine gunner has his mask on. It's an open cockpit.
Chosen answer: The Core didn't originate the name - it's been used since the 50's and even has its own Wikipedia article at http://en.wikipedia.org/wiki/Unobtainium. There it's described as "any fictional, extremely rare, costly, or impossible material, or (less commonly) device needed to fulfill a given design for a given application."