From your title:
register_sidebar_widget()is deprecated since version 2.8! Use wp_register_sidebar_widget() instead.
Somewhere in your Theme – in functions.php, or in a function included within functions.php – you’re adding custom Widgets. Look for that code, and replace usage of register_sidebar_widget() with wp_register_sidebar_widget().
PROTIP: Use the Log Deprecated Notices Plugin to track down where the deprecated function call is being executed.