Displaying custom menu with menu descriptions and icons in a WordPress widget. http://sridharkatakam.com/display-custom-menu-menu-descriptions-icons-wordpress-widget/

functions.php

//* Add walker class that displays menu item descriptions
class Menu_With_Description extends Walker_Nav_Menu {
	function start_el(&$output, $item, $depth, $args) {
		global $wp_query;
		$indent = ( $depth ) ? str_repeat( "\t", $depth ) : '';

		$class_names = $value = '';

		$classes = empty( $item->classes ) ? array() : (array) $item->classes;

		$class_names = join( ' ', apply_filters( 'nav_menu_css_class', array_filter( $classes ), $item ) );
		$class_names = ' class="' . esc_attr( $class_names ) . '"';

		$output .= $indent . '<li id="menu-item-'. $item->ID . '"' . $value . $class_names .'>';

		$attributes = ! empty( $item->attr_title ) ? ' title="' . esc_attr( $item->attr_title ) .'"' : '';
		$attributes .= ! empty( $item->target ) ? ' target="' . esc_attr( $item->target ) .'"' : '';
		$attributes .= ! empty( $item->xfn ) ? ' rel="' . esc_attr( $item->xfn ) .'"' : '';
		$attributes .= ! empty( $item->url ) ? ' href="' . esc_attr( $item->url ) .'"' : '';

		$item_output = $args->before;
		$item_output .= '<a'. $attributes .'>';
		$item_output .= $args->link_before . apply_filters( 'the_title', $item->title, $item->ID ) . $args->link_after;
		$item_output .= '<br /><span class="sub">' . $item->description . '</span>';
		$item_output .= '</a>';
		$item_output .= $args->after;

		$output .= apply_filters( 'walker_nav_menu_start_el', $item_output, $item, $depth, $args );
	}
}

//* Make Font Awesome available
add_action( 'wp_enqueue_scripts', 'enqueue_font_awesome' );
function enqueue_font_awesome() {

	wp_enqueue_style( 'font-awesome', '//netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css' );

}

gistfile1.php

<?php $walker = new Menu_With_Description; ?>

<?php wp_nav_menu( array( 'menu' => 'Side Menu', 'menu_class' => 'menu', 'walker' => $walker ) ); ?>

style.css

/* WP menu with menu descriptions and icons in a widget
------------------------------------------------------------- */

#menu-side-menu li {
	margin-bottom: 20px;
	position: relative;
}
#menu-side-menu li:last-child {
	margin-bottom: 0;
}

#menu-side-menu a {
	font-weight: bold;
	border-bottom: none;
	display: block;
	padding: 10px 10px 10px 65px;
	line-height: 1.3;
	border-radius: 3px;
}

#menu-side-menu a .sub {
	font-weight: normal;
}

#menu-side-menu .current-menu-item a, #menu-side-menu a:hover {
	background: #f1f9f7;
	color: #000;
}

#menu-side-menu a:before {
	font-family: FontAwesome;
	font-size: 40px;
	float: left;
	font-style: normal;
	font-weight: normal;
	text-decoration: inherit;
	padding-right: 15px;
	position: absolute;
	top: 5px;
	left: 10px;
}

#menu-side-menu .menu-item-home a:before {
	content: "\f015";
}

#menu-item-71 a:before {
	content: "\f0a1";
}

#menu-item-72 a:before {
	content: "\f044";
}

#menu-item-73 a:before {
	content: "\f02d";
}

#menu-item-74 a:before {
	content: "\f003";
}

#menu-item-76 a:before {
	content: "\f07a";
}