What should I name my overriding render-function?
It depends on what you want to target.

If it's the tree of the menu you can use theme_menu_tree :

function drupal_subtheme_menu_tree__menu__menu_usm_navbar_small($variables) {
   return '&lt;ul class="menu"&gt;' . $variables['tree'] . '&lt;/ul&gt;';
}


if it's the link, you can use theme_menu_link :

function drupal_subtheme_menu_link__menu__menu_usm_navbar_small($variables) {
   $element = $variables['element'];
   $sub_menu = '';

   if ($element['#below']) {
     $sub_menu = drupal_render($element['#below']);
   }
   $output = l($element['#title'], $element['#href'], $element['#localized_options']);
   return '&lt;li' . drupal_attributes($element['#attributes']) . '&gt;' . $output . $sub_menu . "&lt;/li&gt;\n";
}


These functions need to be in template.php