General:
* Cliché
* Platitude
* Banality
* Trite saying
* Commonplace remark
* Overused phrase
* Stock phrase
* Repetitive statement
* Unoriginal remark
More specific:
* Comforting phrase (if the bromide is intended to soothe or reassure)
* Wisecrack (if the bromide is intended to be witty or humorous)
* Slogan (if the bromide is a catchphrase or motto)
* Proverb (if the bromide is a traditional saying)
The best synonym will depend on the context of the sentence and the tone you want to convey.