When telling someone to "button your lip," you're basically asking them to be quiet or stop talking. However, there are several synonyms you can use for this phrase, depending on the context or situation.
For instance, "put a sock in it" or "zip your lip" can be used instead of "button your lip." These phrases can be slightly humorous and are perfect when you're trying to lighten the mood.
If you want to be more direct, you can use "shut up!" or "stop talking!" These phrases are more forceful and are best reserved for situations where the person you're addressing is being disruptive or disrespectful.
In any case, the goal is for the person to stop talking, so choose the synonym that suits the situation and your desired tone.