The name Quantum3 is taken from the internal technology High Voltage Software uses within the engine. The number "3" at the end refers to the generation of the engine, this being the third generation Quantum engine. 
This engine is not Wii specific, as it is also used in games for the PlayStation 2, PlayStation Portable, Nintendo 3DS and Direct X 9-equipped PCs. It allows cross platform data sharing not only limited to character models, but includes a wide variety of data sharing features such as maps, region, UI, navigation data, collision data, movies, audio and also includes game logic. 
At any time the engine can be improved on as features and platform support can be added and removed from the engine its self.
With Wii specifics, the Quantum3 engine allows bump mapping, textures and lighting that are not normally capable on the Wii. In the processing of building on the Wii's unique technical hardware, HVS has created some new tricks and techniques such as Dynametric Light Tightening, and Reframbriance which takes advantage of the Wii's unique hardware and allow maximum flexability.