A bromide is a phrase or platitude whose excessive use suggests insincerity or a lack of originality in the speaker.
The term "bromide" derives from the antiquated use of bromide salts in medicine as mild tranquilizers and sedatives. Administration of a "bromide" (such as the original Bromo-Seltzer before 1975 in the US) would relieve anxiety and make the patient drowsy.
Describing a phrase as a "bromide" is meant humorously to imply that it is a verbal sedative, a boring statement with similar soporific properties.
In 1906, the author Gelett Burgess published a book called Are You a Bromide? in which he referred to boring people as "bromides".