need to modify core file object method WP_Widget_Pages->widget

What are you actually trying to achieve and why? If we know we can maybe help you, or at least point you in the right direction.

Those values are speciied per sidebar and can be set differently for each sidebar –

$args = array(
	'name'          => sprintf(__('Sidebar %d'), $i ),
	'id'            => 'sidebar-$i',
	'description'   => '',
	'before_widget' => '<li id="%1$s" class="widget %2$s">',
	'after_widget'  => '</li>',
	'before_title'  => '<h2 class="widgettitle">',
	'after_title'   => '</h2>' ); ?>
register_sidebar( $args );

You best and easiest bet would be to create sidebars using the above code in the ‘functions.php’ file. That way, they can be updated dynamically and you will never need to re-hack.

Look here for further information – Function Reference/register sidebar

You can also make your own custom widgets that will achieve everything you want. A live example that is used on my companies website (http://www.dynedrewett.com) is below.

Could you also achieve what you want through looking at the scource and amending a class that is already used in the ‘UL’ container? This would perhaps save rewriting all of the widgets.

Recommended reading is Multi-widgets

The code for the default widgets is in ‘wp_includes\default-widgets.php’ – you could copy any you want in to a ‘custom-widgets.php’ file and amend as necessary.

/**
 * Declare your custom multi-widget
 */
class Custom_Widget_Brochures extends WP_Widget {
    /** constructor */
    function Custom_Widget_Brochures() {
		$widget_options = array('classname' => 'custom_widget_brochures', 'description' => __('Includes a list of available Brocures in the sidebar.'));
        parent::WP_Widget(false, $name = 'Brochures', $widget_options);
    }

    /** @see WP_Widget::widget */
    function widget($args, $instance) {
        extract($args);
		$title = apply_filters('widget_title', $instance['title']);
		if(!$title || $title === '' || $title === null)
			$title = 'View our brochures';
        ?>
		<?php echo $before_widget; ?>
		<?php
		$brochures = get_posts('numberposts=10&post_type='.POST_TYPE_BROCHURE.'&orderby=name&order=ASC');
		if(!empty($brochures)){
		?>
			<div id="sidebar-brochures" class="rightColItem rc_title_bar rc_title">
				<?php echo $title; ?>
			</div>
			<div class="rightColItem pinkLinks">
				<ul class="rc_linkList">
					<?php foreach($brochures as $brochure){ ?>
						<li class="link_arrow"><a href="<?php echo get_permalink($brochure->ID); ?>"><?php echo $brochure->post_title; ?></a></li>
					<?php } ?>
				</ul>
			</div>
		<?php } ?>
		<?php echo $after_widget; ?>
		<?php
    }

	/** @see WP_Widget::update */
    function update($new_instance, $old_instance){
	$instance = $old_instance;
	$instance['title'] = strip_tags($new_instance['title']);
        return $instance;
    }

    /** @see WP_Widget::form */
    function form($instance){
		$title = esc_attr($instance['title']);
        ?>
			<p>
				<label for="<?php echo $this->get_field_id('title'); ?>">
					<?php _e('<strong>Title:</strong>'); ?>
					<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; ?>" />
				</label>
			</p>
        <?php
    }
}

/**
 * Register all of your custom WordPress widgets on startup
 */
add_action('widgets_init', 'custom_widgets_init');
function custom_widgets_init() {
	register_widget('Custom_Widget_Brochures');
}

/**
 * Remove unwanted default sidebar widgets
 */
add_action('admin_head', 'edit_registered_widgets');
function edit_registered_widgets(){
	global $wp_registered_widgets;
	unset($wp_registered_widgets['pages-1']);
	unset($wp_registered_widgets['calendar-1']);
	unset($wp_registered_widgets['archives-1']);
	unset($wp_registered_widgets['links-1']);
	unset($wp_registered_widgets['meta-1']);
	unset($wp_registered_widgets['search-1']);
	unset($wp_registered_widgets['text-1']);
	unset($wp_registered_widgets['categories-1']);
	unset($wp_registered_widgets['recent-posts-1']);
	unset($wp_registered_widgets['recent-comments-1']);
	unset($wp_registered_widgets['rss-1']);
	unset($wp_registered_widgets['tag_cloud-1']);
	unset($wp_registered_widgets['nav_menu-1']);
}

Hope that helps.

You’re welcome. I was also very pleased when I discovered how to do this, as it really is a powerful weapon in WP’s arsenal!