1
general::
Exactly right, precisely, as in Her review of the book was right on the button, or We're supposed to be there at six o'clock on the button. This term may come from boxing, where a punch on the button means "a punch on the chin." [Colloquial; c. 1900] Also see ON THE DOT.
American Heritage Idioms
2
general::
Phrase(s): on the button
exactly right; in exactly the right place; at exactly the right time. • That’s it! You’re right on the button. • He got here at one o’clock on the button.
McGrawhill's American Idioms And Phrasal Verbs