The name — often also spelled as unobtanium — combines unobtainable with the -ium suffix that marks the names of chemical elements.
It goes back a long way. This is how it was defined in its first appearance in print:
A substance having the exact high test properties required for a piece of hardware or other item of use, but not obtainable whether because it theoretically cannot exist or because technology is insufficiently advanced to produce it.
Interim Glossary, Aero-Space Terms, by Woodford Heflin, Feb. 1958, reproduced in Paul Dickson’s A Dictionary of the Space Age, 2009. The glossary was issued by the Air University of the US Air Force. Heflin tagged the word as “humorous or ironical”.
You can see how the engineers in aeronautics and the space program, who were forced to invent novel materials to withstand extreme temperatures, would have had a special need for such a word. Unobtainium has long since become almost a standard term in engineering circles for a material that would instantly solve a tricky design problem, if only it existed.
It can also be some substance that does exist but is too difficult to make or get hold of to be practical; another recent term for such stuff is unaffordium. It also turns up a lot in this sense in sports such as motor racing and mountain biking for highly desirable products that are at the most expensive end of their range:
Besides the state-of-the-art 249cc two-stroke motor that delivered the explosive power that motocross required, the bike was loaded with unobtainium parts, including magnesium hubs, electronic ignition and reed-valve induction.
American Motorcyclist, Oct. 2005.
The term has become much better known among non-specialists following two recent SF films. In The Core of 2003, a mad scientist named Edward Brazzleton invents a metal by that name to withstand the extraordinary temperatures and pressures at the Earth’s core. James Cameron’s Avatar (2009) is set on the distant world Pandora, home to the Na’vi, where humans are mining for the rare mineral unobtainium.
Imaginatively exotic materials whose properties usefully disregard the boring laws of physics have long been features of SF writing, though authors usually prefer to create their own names for them, such as cavorite, kryptonite, scrith, dilithium and carbonite. The term is so well known as an umbrella tag for such materials (along with its close relatives impossibilium and handwavium) that it’s usually employed knowingly or tongue-in-cheek when it does appear:
None of the other moons in the Kthsemenee system had the one attribute this one possessed: a core of almost one percent unobtainium.
Startide Rising, by David Brin, 1983.