Randomizing the RSS Widget

Here’s one way to achieve this:

/**
 * Display a random feed in the RSS Widget
 * Activated by the wpse_random_url string in the feed url
 *
 * @see http://wordpress.stackexchange.com/a/187599/26350
 */

! is_admin() && add_filter( 'wp_feed_options', function( $feed, $url )
{
    // Modify this list of feeds to your needs:
    $urls = [
        'http://stackoverflow.com/feeds/', 
        'http://wordpress.stackexchange.com/feeds',
        'http://unix.stackexchange.com/feeds'
    ];

    // Select a random feed from the above list:            
    if(    class_exists( '\SimplePie' )
        && $feed instanceof \SimplePie 
        && false !== strpos( $url, 'wpse_random_url' ) 
        && method_exists( $feed, 'set_feed_url' ) 
    )
            $feed->set_feed_url( $urls[ array_rand( $urls ) ] );

    return $feed;
}, 10, 2 );

where our feed url must contain the wpse_random_url string.

We could for example use our current site’s feed, with the random_url GET parameter:

http://example.tld/feed/?wpse_random_url

and add that to the url field of the RSS widget:

Random feed GET parameter

The next step to improve this mini plugin, would be to introduce a way to add the list of feeds, directly from the backend instead of having to modify it within the code itself.