WordPress has a mechanism exactly for this: actions and filters.
BuddyPress leverages actions/hooks to provide extension points through which you can customize it without modifying it's sources.
For what you need there's a do_action( 'bp_core_activated_user', $user_id, $key, $user )
hook defined, so you'd just have to register add_action( 'bp_core_activated_user', 'bp_core_new_user_activity' )
that will be called with $user_id, $key, $user
as parameters.
If you search the documentation you'll find most of the hooks. But ideally you'd look at the sources themselves to find appropriate ones.