Logically the other side of white is not white, while the antonym is the extreme black.
By and by, which was once a synonym, has become an antonym of immediately, meaning at some (perhaps remote) future time.
In strict usage, daily is the antonym of nightly as diurnal is of nocturnal.
1867, coined to serve as opposite of synonym, from Greek anti- "equal to, instead of, opposite" (see anti-) + -onym "name" (see name (n.)). Perhaps introduced to English in the book "Synonyms and Antonyms" (1867) by the Ven. C.J. Smith, M.A.
UNDER the head of Synonyms and Antonyms, Archdeacon Smith arranges words which form an antithesis to one another. The word "antonym" is, we believe, a new formation but useful. ["Journal of Sacred Literature," July 1867]French antonyme (1842), German antonym (by 1859) are older. The un-Greek alternative counterterm has been left to fade.