Display Menu Name using wp_nav_menu

If you know the menu’s slug, then things are easier, otherwise you can use this function to get the menu at a specified location.

<?php
function wpse45700_get_menu_by_location( $location ) {
    if( empty($location) ) return false;

    $locations = get_nav_menu_locations();
    if( ! isset( $locations[$location] ) ) return false;

    $menu_obj = get_term( $locations[$location], 'nav_menu' );

    return $menu_obj;
}
?>

Then

//if you after the menu the menu with a specific ID / Slug
//$menu_obj =wp_get_nav_menu_object($id_slug_or_name); 

//if you after the menu at a specific location
$menu_obj = wpse45700_get_menu_by_location($location); 

echo "<h3>".esc_html($menu_obj->name)."</h3>";
//Display menu here

Or, rather than echo the html, you could pass it as part of the the argument for the items attribute in wp_nav_menu.

For example, to display the menu at location ‘primary’:

$location = 'primary';
$menu_obj = wpse45700_get_menu_by_location($location ); 
wp_nav_menu( array('theme_location' => $location, 'items_wrap'=> '<h3>'.esc_html($menu_obj->name).'</h3><ul id=\"%1$s\" class=\"%2$s\">%3$s</ul>') ); 

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