There are a number of synonyms for the phrase "have a big mouth," which generally refers to someone who speaks too loudly, too often, or without thinking before speaking. Some possible alternatives might include phrases like "talk too much," "open one's mouth too often," or simply "be overly talkative." Other options might include terms like "gossip," "blabbermouth," or "loudmouth," which emphasize the idea that the person in question is spreading information that they shouldn't be. Ultimately, all of these phrases speak to the idea that the person in question needs to learn to think before speaking, and to keep some things to themselves.