Where do I add a new table’s name in wpdb?

$wpdb->tables[] = 'friends'; is the basic code you’re looking for.

Please note that you’ll want to test for the existence of the table in the list so you don’t wind up with 100 copies of the table in the tables array, especially if there’s a chance your code could be repeated within a page load.

UPDATED with example:

add_action( 'wp_loaded', 'add_table' );

function add_table() {
    global $wpdb;
    if ( ! in_array( 'friends', $wpdb->tables ) ) {
         $wpdb->tables[] = 'friends';
    }
}