Get Random Posts with Recently

functions.php

<?php
// https://github.com/cabrerahector/recently/

/**
 * Have Recently return random posts
 * for widget instance ID 'recently-2'
 *
 * @param  array   $args       The args passed to WP_Query.
 * @param  string  $widget_id  The ID of the current widget instance.
 * @return array
 */
function recently_get_random_posts( $args, $widget_id ){
    if ( 'recently-2' == $widget_id )
        $args['orderby'] = 'rand';

    return $args;
}
add_filter( 'recently_pre_get_posts', 'recently_get_random_posts', 10, 2 );

Replace the string recently-2 with the actual instance ID of the widget you want to manipulate. You can use Get Widget ID to easily identify the ID of each widget.