How can I edit the Woocommerce product widget html class?

You can use the given filter to change the html.

Use this in your themes functions.php file

function wpse240457_add_class($html) {
  $html="<ul class="product_list_widget your-new-class">";
  return $html;
}
add_filter('woocommerce_before_widget_product_list', 'wpse240457_add_class', 1, 15);