Editing an existing widget.

There are no template files for widgets. Do you want the html in the menu itself or underneath it? Maybe just hardcode in the template itself.

I want it in the menu itself (within the generated <aside> tags).

But all that I see in the template is <?php dynamic_sidebar( ‘sidebar-3’ ); ?>. So there is nowhere to insert code.

There is no easy way to do this.
How does your theme register sidebars. Look for something similar to this in your theme’s functions.php file:

register_sidebar( array(
			'name' => __( 'Main Sidebar', 'twentytwelve' ),
			'id' => 'sidebar-1',
			'description' => __( 'Appears on posts and pages except the optional Front Page template, which has its own widgets', 'twentytwelve' ),
			'before_widget' => '<aside id="%1$s" class="widget %2$s">',
			'after_widget' => '</aside>',
			'before_title' => '<h3 class="widget-title">',
			'after_title' => '</h3>',
		) );

This is my custom sidebar in my child functions.php

<?php

function twentythirteenchild_widgets_init() {

	register_sidebar( array(
		'name'          => __( 'Third Widget Area', 'twentythirteen' ),
		'id'            => 'sidebar-3',
		'description'   => __( 'Is only for the front page.', 'twentythirteen' ),
		'before_widget' => '<aside id="%1$s" class="widget widget-fp1">',
		'after_widget'  => '</aside>',
		'before_title'  => '<h3 class="widget-title">',
		'after_title'   => '</h3>',
	) );
}

add_action( 'widgets_init', 'twentythirteenchild_widgets_init' );

?>

Wow… I could just us the after_widget attribute. Oh, no that would add it to every widget, I only want to edit that particular one.

By putting this in your theme’s functions.php you can add it after all nav menu widgets:

add_filter('dynamic_sidebar_params', 'add_classes_to__widget');
function add_classes_to__widget($params){

    if (substr($params[0]['widget_id'], 0, 8) == "nav_menu"){ //make sure its a nav menu

        $params[0]['after_widget'] .= '<h1>My html here</h1></aside>';
    }
    return $params;
}