This script add sidebar menu widget at your WordPress theme.

sidebar_menu_widget.php

<?php
class Dvault_Sidebar_Menu extends WP_Widget {

	//setup the widget name, description, etc...
		public function __construct() {

			$widget_ops = array(
				'classname' => 'dvault-sidebar-menu-widget',
				'description' => 'Add a custom menu in sidebar. ',
				);
			parent::__construct( 'dvault-sidebar-menu', 'Dvault Sidebar Menu', $widget_ops );
		}
	//back-end display of widget
		public function form( $instance ) {
      $title = isset($instance['title']) ? $instance['title']: '';
			$nav_menu = isset( $instance['nav_menu'] ) ? $instance['nav_menu'] : '';
			$menus = wp_get_nav_menus();
			?>
      <p>
        <label for="<?php echo $this->get_field_id('title'); ?>">Title:</label></br>
        <input class="widefat" id="<?php echo $this->get_field_id('title'); ?>" type="text" name="<?php echo $this->get_field_name('title'); ?>" value="<?php echo $title; ?>">
      </p>
      <p>
				<label for="<?php echo $this->get_field_id( 'nav_menu' ); ?>"><?php _e( 'Select Menu:' ); ?></label>
				<select id="<?php echo $this->get_field_id( 'nav_menu' ); ?>" name="<?php echo $this->get_field_name( 'nav_menu' ); ?>">
					<option value="0"><?php _e( '&mdash; Select &mdash;' ); ?></option>
					<?php foreach ( $menus as $menu ) : ?>
						<option value="<?php echo esc_attr( $menu->term_id ); ?>" <?php selected( $nav_menu, $menu->term_id ); ?>>
							<?php echo esc_html( $menu->name ); ?>
						</option>
					<?php endforeach; ?>
				</select>
			</p>
			<?php
		}

		function update( $new_instance, $old_instance ) {
			$instance = $old_instance;
      $instance['title'] = $new_instance['title'];
      $instance['nav_menu'] = $new_instance['nav_menu'];

			return $instance;
		}

	//front-end display of widget
		public function widget( $args, $instance ){
			$pageId = get_the_ID();
			$currentClass = '';
			$title = $instance['title'];
			$nav_id = $instance['nav_menu'];
			$nav_menu = ! empty( $instance['nav_menu'] ) ? wp_get_nav_menu_object( $instance['nav_menu'] ) : false;
			$wp_menus = wp_get_nav_menu_items($instance['nav_menu']);
      echo $args['before_widget'];
		 ?>
		 <div class="widget-title"><?php echo $title; ?></div>
	   <div class="menu-<?php echo $nav_menu->slug; ?>-posts-container">
	     <ul id="menu-<?php echo $nav_menu->slug; ?>-posts" class="menu">
				 <?php foreach ($wp_menus as $wp_menu) :
					 $thumbImage = wp_get_attachment_url(get_post_thumbnail_id($wp_menu->object_id));
					 $thumbImage = (isset($thumbImage) && $thumbImage != '') ? $thumbImage : '';
					 $pageIDFromMenu = get_post_meta( $wp_menu->ID, '_menu_item_object_id', true );
					 $pageIDFromMenu;
					 if( $pageId == $pageIDFromMenu){
						 $currentClass = 'current-menu-item';
					 } else {
						 $currentClass = '';
					 }
					 ?>
	       <li id="menu-item-<?php echo $wp_menu->ID; ?>" class="menu-item menu-item-type-<?php echo $wp_menu->type; ?> menu-item-object-<?php echo $wp_menu->object; ?> menu-item-<?php echo $wp_menu->ID; ?> <?php echo $currentClass; ?>">
	         <a href="<?php echo $wp_menu->url; ?>"><?php echo $wp_menu->title; ?></a>
					 <div class="sidebar-submenu-hover">
						 <?php if( !empty($thumbImage) ) : ?>
						 <img src="<?php echo $thumbImage; ?>" alt="" class="img-responsive">
					 <?php endif; ?>
						 <a class="sidebar-submenu-hover-title"><?php echo $wp_menu->title;  ?></a>
					 </div>
	       </li>
			 <?php endforeach; ?>
	 </ul></div>
			<?php
    		echo $args['after_widget'];
    }
  }

/*
============================
widget action hook
============================
*/
add_action( 'widgets_init', function() {
	register_widget( 'Dvault_Sidebar_Menu' );
} );