Introduce  BuddyPress Private Message:

BuddyPress Private Message Rate Limiter plugin allows you to limit users from sending large number of messages and flooding other users on your BuddyPress based social network. The best use case for this plugin is to stop spammers from flooding your users private message.

• Site Admin can limit only the new messages (best suited for spam prevention)
• Site Admins can limit the new messages and replies both ( We recommend the 1st option to avoid spammers but let regular users continue)
• Site Admin can set the duration of time for which the limit will apply