Unix was named Unics after Multics, the project that the Unix developers had worked on before becoming frustrated with its complexity and delays.
Originally called the “unnamed PDP-7 operating system”, computer scientist Brian Kernighan recalls: "I suggested, based on Latin roots, that since Multics provided many of everything and the new system had at most one of anything, it should be called “UNICS”, a play on ‘uni’ instead of ‘multi’.”
An alternative memory is that Peter Neumann came up with UNICS as an acronym for Uniplexed Information and Computing Service and that 'Unics' was intended to be pronounced “eunuchs”, as a play on the fact that it was a castrated version of Multics. There is perhaps some truth in this because apparently their legal team made them change the spelling to Unix.
Sources: Emails from Ken Thompson and Brian Kernighan | Oxford English Dictionary