Continuity mistake: In the scene where Parker is shown playing golf on the indoor golf course in RDA, the first time he hits the ball, there are two other balls lying on the course side by side. In the next shot, two are distant apart and in the shot after that, they are in their original position. (00:12:15)
Continuity mistake: When Jake first gets into the "syncing" device and he says, "This is cool," Grace's hands are already on his leg. In the next shot she reaches for his legs, where he replies, "Don't, I got this."
Continuity mistake: AMP suit details such as the windshield wiper blade/arm as well as the grab handles on top of the cockpit appear and disappear throughout the film, depending on whether a CG model or practical prop are being used.
Continuity mistake: In a scene in the planning room (where the computer model of Pandora is being studied by Colonel Quaritch and Parker Selfridge), Jake is watching them move the model quickly to find something on it and they show his digital watch which says 2:19. Less than a minute later they show his watch again and it says 2:51.
Continuity mistake: In the scene after Jake was just in his avatar for the first time, he and the Colonel are discussing things around a holographic 3D map. With each shot, Jake's arms change from being straight out holding the edge of the table, to clasped in front of him, with no time to move the position in between. (00:49:30)
Continuity mistake: When Jake is walking up to the Tree of Souls to ask Eywa for help, he kneels and reaches for his braid as if it were hanging over his shoulder when it is lying down his back. The angle changes and it is now over his shoulder. When he stands up as Neytiri tells him that Eywa protects the balance of life, it's suddenly down his back again.
Continuity mistake: When Eytukan orders Jake and Grace to be tied up, Tsu'tey and another Na'vi grab Jake and bind his wrists, his right on top. It switches to Eytukan shouting a war cry, and goes back to Jake; his left wrist is now on top.
Continuity mistake: After Grace has been shot, when she and Jake are at the remote link-up site, Jake pulls a blanket up to Grace's shoulders, and the tip of it is still folded over. When the shot changes to a close-up of Grace, the blanket now lies completely flat. (01:52:30)
Continuity mistake: When Jake's avatar is running through the garden as he first enjoys having working legs again, he skids to a stop on the loose topsoil in the aisle of the garden. The next shot that shows him wriggling his toes in the soil reveals a pristine path of soil and no evidence of the disturbance his skidding to a stop caused.
Continuity mistake: (Extended Edition) In the scene we see Tsu'tey badly injured on the ground, he asks Jake to finish him off in the "Navi way". Tsu'tey reaches out for Jake's dagger and Jake assists him in taking the dagger out of the scabbard. In the next shot the dagger is back in Jake's scabbard and a moment later in the third shot the dagger is now out of the scabbard and in Jake's hand. (01:07:05)
Continuity mistake: There is a scene where Jake and Grace are tied up by the natives, shortly after they are cut down, Jake pushes Grace over a log to avoid an explosion, the shot changes to something else, then returning to the log, and we see that he's pushing her again. then the shot continues.
Continuity mistake: When Selfridge shows Grace the unobtainium ore, he picks it up with his left hand. In the next shot we can briefly see him passing the ore from his right hand into his left hand. The ore magically changed hands between the shots. (00:13:45)
Continuity mistake: In the colonel's robot there is a rear-view mirror. When we see the robot from the outside, it is very close to his head. But when we see shots from inside the robot, there is plenty of room around his head and we don't see the mirror.
Continuity mistake: In the scene where Jake wakes up from his link and Grace asks him, "Is the avatar safe?" and he replies, "Yeah Doc, and you are not gonna believe where I am," her hand is on top of his shoulder. But as the camera angle changes back and forth her hand switches to the side of his upper arm and back to the shoulder without time to have made the move. (00:48:55)
Continuity mistake: When Jake tames Toruk, he confronts the Na'vi and speaks to Tsutey. While Jake is speaking to Tsutey, Tsutey's braids are over his shoulder. After Jake says "You are Oloektan," all but a single braid are behind his shoulder. After this, when Tsutey looks at the Toruk in the distance, a braid or two fall over his shoulder, in the next cut no braids are over his shoulder.
Continuity mistake: When Selfridge is on the indoor putting green and says "Grace, you know, I enjoy our little talks", a woman walks behind him. A few seconds later the same woman walks behind him again, in the same direction. (00:12:40)
Continuity mistake: In the scene where Neytiri is showing Seze to Jake, Neytiri's ponytail changes positions between shots twice. She puts it over her left shoulder as Seze is landing, it's at her back in the next shot of her, then when she says "Ikran is not horse" it's over her shoulder again. (00:58:15)
Continuity mistake: When Sigourney Weaver is shot in the stomach, she is not wearing make-up. When she is transported to the Tree of Life and laid under it for possible rejuvenation, she is wearing mascara.
Continuity mistake: When Quaritch is scolding Jake for making out with a Na'vi, Jake moves his head to the right. In the immediate next angle, his head is straight. (01:31:25)
Continuity mistake: When Jake is learning to ride on the direhorse, after he is corrected when calling she a he, and is hit on the head by the direhorse's queue, his braid slips over his right shoulder, but next shot the braid has returned to his back.
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."