Theme only Custom Widget in functions.php -- this example modifies the page widget to display only a page's children links

gistfile1.php

class Custom_Widget_Pages extends WP_Widget_Pages {

  function Custom_Widget_Pages() {
    $widget_ops = array('classname' => 'widget_pages', 'description' => __( "Display a page's children") );
    $this->WP_Widget('pages', __('Page Children Only'), $widget_ops);
  }

  function widget( $args, $instance ) {
    extract( $args );

    $post_id =  get_queried_object_id();

    $title = apply_filters('widget_title', empty( $instance['title'] ) ? __( 'Page Children Only' ) : $instance['title'], $instance, $this->id_base);
    $sortby = empty( $instance['sortby'] ) ? 'menu_order' : $instance['sortby'];
    $exclude = empty( $instance['exclude'] ) ? '' : $instance['exclude'];

    if ( $sortby == 'menu_order' )
      $sortby = 'menu_order, post_title';

    $out = wp_list_pages(
            apply_filters('widget_pages_args',
              array('title_li' => '', 'echo' => 0, 'child_of' => $post_id, 'depth'=> 1, 'sort_column' => $sortby, 'exclude' => $exclude) ) );

    if ( !empty( $out ) ) {
      echo $before_widget;
      if ( $title )
        echo $before_title . $title . $after_title;
    ?>
    <ul>
      <?php echo $out; ?>
    </ul>
    <?php
      echo $after_widget;
    }
  }

  function update( $new_instance, $old_instance ) {
    $instance = $old_instance;
    $instance['title'] = strip_tags($new_instance['title']);
    if ( in_array( $new_instance['sortby'], array( 'post_title', 'menu_order', 'ID' ) ) ) {
      $instance['sortby'] = $new_instance['sortby'];
    } else {
      $instance['sortby'] = 'menu_order';
    }

    $instance['exclude'] = strip_tags( $new_instance['exclude'] );

    return $instance;
  }

  function form( $instance ) {
    //Defaults
    $instance = wp_parse_args( (array) $instance, array( 'sortby' => 'post_title', 'title' => '', 'exclude' => '') );
    $title = esc_attr( $instance['title'] );
    $exclude = esc_attr( $instance['exclude'] );
  ?>
    <p><label for="<?php echo $this->get_field_id('title'); ?>"><?php _e('Title:'); ?></label> <input class="widefat" id="<?php echo $this->get_field_id('title'); ?>" name="<?php echo $this->get_field_name('title'); ?>" type="text" value="<?php echo $title; ?>" /></p>
    <p>
      <label for="<?php echo $this->get_field_id('sortby'); ?>"><?php _e( 'Sort by:' ); ?></label>
      <select name="<?php echo $this->get_field_name('sortby'); ?>" id="<?php echo $this->get_field_id('sortby'); ?>" class="widefat">
        <option value="post_title"<?php selected( $instance['sortby'], 'post_title' ); ?>><?php _e('Page title'); ?></option>
        <option value="menu_order"<?php selected( $instance['sortby'], 'menu_order' ); ?>><?php _e('Page order'); ?></option>
        <option value="ID"<?php selected( $instance['sortby'], 'ID' ); ?>><?php _e( 'Page ID' ); ?></option>
      </select>
    </p>
    <p>
      <label for="<?php echo $this->get_field_id('exclude'); ?>"><?php _e( 'Exclude:' ); ?></label> <input type="text" value="<?php echo $exclude; ?>" name="<?php echo $this->get_field_name('exclude'); ?>" id="<?php echo $this->get_field_id('exclude'); ?>" class="widefat" />
      <br />
      <small><?php _e( 'Page IDs, separated by commas.' ); ?></small>
    </p>
  <?php
  }
}

add_action('widgets_init',function(){
     return register_widget('Custom_Widget_Pages');
});