Robin Waterfield, in the introduction to his edition of Republic
(published under the Oxford World's Classics series, if you're looking for it), writes:
The title Republic is a bad translation of the Greek politeia. The Greek word does occur a number of times in the book, as well as forming the title [...]. Politeia is the public and political life of a community; in Latin this is res publica, 'public business'; Greek works used to be referred to by their Latin or Latinized titles: hence Republic. (pp. xi-xii)
If I'm reading the academic material correctly, the name ΠΟΛΙΤΕΙΑ was given to the work by Plato himself. Since classicists are traditionally perverse enough to write their commentaries and annotations to Greek texts in Latin, the title Res Publica
(which, at least by Latin standards, is a fairly sensible rendering of the Greek) gained currency, and from there became the English Republic
— which, unfortunately, has gained a fair amount of meaning on its own that isn't present in the original Latin or Greek words.