My best guess is that at a particularly cold football game in the Palouse, one of the students may have had a bit too much Peppermint Schnapps in their hot chocolate and came across "W.S.U." while reading the program and instead of recognizing it at an acronym for Washington State University tried to pronounce it as a word and "Wazzu" is what ended up coming out. I guess it stuck.

Go Cougs!