The Bell Labs documentation was very careful to always spell UNIX in all capital letters, and small caps whereever possible.
The name pokes fun at a mainframe operating system called MULTICS. The authors of UNIX thought MULTICS was far too complicated for what it does. Excessive complexity is considered a sign of poor software design, and indicates poor software design management.
I believe Linux (the whole phenomenon) is fairly close to what the authors of UNIX intended UNIX to become.