Custom maintenance page

When WordPress goes into maintenance mode, it adds a file named .maintenance to the root directory while the maintenance is being performed, then it’s removed afterwards. You can write a function inside your theme’s functions.php that checks for this file and loads a custom maintenance page from the theme.

if ( ! function_exists( 'wpse84987_maintenance_mode' ) ) {
    function wpse84987_maintenance_mode() {
        if ( file_exists( ABSPATH . '.maintenance' ) ) {
            include_once get_stylesheet_directory() . '/maintenance.php';
            die();
        }
    }
    add_action( 'wp', 'wpse84987_maintenance_mode' );
}

Put your maintenance content in the maintenance.php page inside your theme folder and you’re all set to style it however you would like.

If you use the wp_die function, you’ll get the standard white box on grey background. This way lets you style your maintenance page like you would any other theme page.

UPDATE: You can also do this outside the theme by adding maintenance.php to the wp-content directory (or wherever you’ve set WP_CONTENT_DIR to point to) as a drop-in plugin. When WP checks for maintenance mode from inside wp_maintenance() it’ll look for that file and load it if present, or load its own if not. If the site isn’t in maintenance mode, or is in it for more than 10 minutes, ‘maintenance.php’ won’t load even though the site is technically still in maintenance mode. WordPress 4.6 introduces the ‘enable_maintenance_mode’ filter, which can be (ab)used by a tool like wp-cli to force the check for the drop-in and would let you run a CLI command from your maintenance file.