Originally a (slightly politically incorrect) term used in boxing, meaning a caucasian
fighter who was deemed capable of taking the title from its current (black
Sometimes used to refer to video games, specifically to describe a title that is considered to be a potential killer app for a given platform. For example, Metal Gear Solid 2 is the Great White Hope of the Sony Playstation 2 (not that it really needs it now, they've managed to shift a great number of units with scarcely any decent original titles available). Street Fighter II was the (broadly successful) Great White Hope of the Super Nintendo in its battle with the Sega Genesis.
The term does not always indicate a game that is actually a killer app, or even very good - for instance, Black & White was seen by Electronic Arts and Lionhead as the Great White Hope of the PC, but it was, frankly, cobblers. Conversely, Quake was a GWH that delivered in spades. (That sounded even more un-PC. Sorry.)
Basically, a term to describe a game that parties with a vested interest (e.g. games publishers, hardware developers or early adopters/die-hard adherents of a platform) think will solve all their problems.