How to override aggregator_block_view() in the aggregator module?
IF you want to alter the block view you can try hook_block_view_alter

Refer to menu block alter example for an example 

http://api.drupal.org/api/drupal/modules%21menu%21menu.module/function/menu_block_view_alter/7

---UPDATE---

  function hook_aggregator_process($feed) {
      foreach ($feed-&gt;items as $item) {
        mymodule_save($item);
      }
    }


Check out the docs at http://api.drupal.org/api/drupal/modules%21aggregator%21aggregator.api.php/function/hook_aggregator_process/7

Also enable the aggregator settings at admin/config/services/aggregator/settings

----UPDATE-----

Also if you just want to override the item list then you should use themename_preprocess_aggregator_summary_items(&amp;$variables) and check for 'summary_list' this holds the items list. For more details refer to template_preprocess_aggregator_summary_items in the aggregator.pages.inc file.