BuddyPress.org

First you’ll need to add an additional option to the BP dropdown select menu for “activity”.
Check out the “bp_search_form_type_select” filter in bp_search_form_type_select() (located in bp-core-templatetags.php) to add your own.

Then you’ll need to modify the bp_ajax_querystring so it adds the search_terms parameter as Andy has stated above.
Check out the “bp_dtheme_ajax_querystring” filter in the bp-default theme’s /_inc/ajax.php.