Adds a custom way of ordering products, in this example it's 'random' product ordering. This can also be set to be used as default via the Catalog tab in WooCommerce settings.

wc-custom-product-order.php

<?php

add_filter( 'woocommerce_get_catalog_ordering_args', 'custom_woocommerce_get_catalog_ordering_args' );

function custom_woocommerce_get_catalog_ordering_args( $args ) {
  $orderby_value = isset( $_GET['orderby'] ) ? woocommerce_clean( $_GET['orderby'] ) : apply_filters( 'woocommerce_default_catalog_orderby', get_option( 'woocommerce_default_catalog_orderby' ) );

	if ( 'random_list' == $orderby_value ) {
		$args['orderby'] = 'rand';
		$args['order'] = '';
		$args['meta_key'] = '';
	}

	return $args;
}

add_filter( 'woocommerce_default_catalog_orderby_options', 'custom_woocommerce_catalog_orderby' );
add_filter( 'woocommerce_catalog_orderby', 'custom_woocommerce_catalog_orderby' );

function custom_woocommerce_catalog_orderby( $sortby ) {
	$sortby['random_list'] = 'Random';
	return $sortby;
}

The random products snippet isn’t working correctly with pagination, since it resets at every new page.