When MU was founded in 1839, it was originally called "Missouri State University" ("MSU"). If you try to pronounce "MSU" as a word instead of an acronym, it sounds a lot like "Mizzou", and that's how the nick name started.
As "Missouri State" became more and more research-intensive, it eventually dropped the "State" from its name and became "The University of Missouri". After this name change (late 19th century), it became acceptable to abbreviate the university as "UM", but a couple things eventually changed that:
1) In 1963, the University of Missouri SYSTEM was created by adding (or rebranding) colleges in St. Louis (UMSL), Kansas City (UMKC), and Rolla (UMR). The flagship campus in Columbia (i.e., Mizzou) became "UMC". The previous abbreviation, "UM", quickly became ambiguous with the new system and naming convention.
2) Around the same time the University System was created, the Big 8 conference decided that all of its schools would be abbreviated with "U" at the end (for consistency). And to this day, all of them still do.
And that's how "MU" became the official abbreviation.
 In 2004, Southwest Missouri State University received permission from the Missouri state senate to shorten its name to Missouri State University. This created controversy as many Mizzou fans and alumni felt that it stole a part of Mizzou's history. Additionally, SMSU was never a land-grant institution, so the name change made little sense from a historical perspective.
 Washington State University (WSU) has the nick name "Wazzu" which started in a similar way, but hasn't been as widely-adopted by fans, alumni, and the media. What makes "Mizzou" so ideal as a nick name is that looks and sounds very similar to "Missouri", making it unambiguous as to which school it's referring to. (Plus, it doesn't sound like "wazoo".)
 MU is unique in that it's both a national research university (AAU, Carnegie RU/VH) and state land-grant university. Research Universities generally do not have a "State" identifier in their name (e.g., U of Michigan, U of Colorado, U of Washington, U of Florida). Land-grant universities, however, which generally have (or once had) a primary mission to educate the local state populace, typically have "State" in their name (e.g., Michigan State, Colorado State, Washington State, Florida State).
 It's unclear whether this was a legal entity change or made informally to suit the current mission and status of the university at the time (as an up-and-coming research institution). If anyone knows, please send the relevant info to @Mizzou2SEC.