wpquestions.com, ID = 3170

<?php

/**
 * set the ID attribute of nav menu items
 */
add_filter( 'nav_menu_item_id', 'pagenamed_id', 10, 2 );

function pagenamed_id( $id, $item ) {

    if ( $content_page_id = get_post_meta ( $item->ID, '_menu_item_object_id', true ) ) {
        if ( $content_page = get_post ( $content_page_id ) )
            $id = $content_page->post_name;
    }

    return $id;

}

/**
 * set classes on nav menu item
 *
 * @note contains a large portion of the function _wp_menu_item_classes_by_context()
 */
add_filter( 'nav_menu_css_class', 'filter_nav_menu_classes', 10, 2 );

function filter_nav_menu_classes( $classes, $menu_item ) {

    $classes = array();

    global $wp_query;

    $queried_object = $wp_query->get_queried_object();
    $queried_object_id = (int) $wp_query->queried_object_id;

    $active_object = '';
    $active_ancestor_item_ids = array();
    $active_parent_item_ids = array();
    $active_parent_object_ids = array();
    $possible_taxonomy_ancestors = array();
    $possible_object_parents = array();
    $home_page_id = (int) get_option( 'page_for_posts' );

    if ( $wp_query->is_singular && ! empty( $queried_object->post_type ) && ! is_post_type_hierarchical( $queried_object->post_type ) ) {
        foreach ( (array) get_object_taxonomies( $queried_object->post_type ) as $taxonomy ) {
            if ( is_taxonomy_hierarchical( $taxonomy ) ) {
                $term_hierarchy = _get_term_hierarchy( $taxonomy );
                $terms = wp_get_object_terms( $queried_object_id, $taxonomy, array( 'fields' => 'ids' ) );
                if ( is_array( $terms ) ) {
                    $possible_object_parents = array_merge( $possible_object_parents, $terms );
                    $term_to_ancestor = array();
                    foreach ( (array) $term_hierarchy as $anc => $descs ) {
                        foreach ( (array) $descs as $desc )
                            $term_to_ancestor[ $desc ] = $anc;
                    }

                    foreach ( $terms as $desc ) {
                        do {
                            $possible_taxonomy_ancestors[ $taxonomy ][] = $desc;
                            if ( isset( $term_to_ancestor[ $desc ] ) ) {
                                $_desc = $term_to_ancestor[ $desc ];
                                unset( $term_to_ancestor[ $desc ] );
                                $desc = $_desc;
                            } else {
                                $desc = 0;
                            }
                        } while ( ! empty( $desc ) );
                    }
                }
            }
        }
    } elseif ( ! empty( $queried_object->post_type ) && is_post_type_hierarchical( $queried_object->post_type ) ) {
        _get_post_ancestors( $queried_object );
    } elseif ( ! empty( $queried_object->taxonomy ) && is_taxonomy_hierarchical( $queried_object->taxonomy ) ) {
        $term_hierarchy = _get_term_hierarchy( $queried_object->taxonomy );
        $term_to_ancestor = array();
        foreach ( (array) $term_hierarchy as $anc => $descs ) {
            foreach ( (array) $descs as $desc )
                $term_to_ancestor[ $desc ] = $anc;
        }
        $desc = $queried_object->term_id;
        do {
            $possible_taxonomy_ancestors[ $queried_object->taxonomy ][] = $desc;
            if ( isset( $term_to_ancestor[ $desc ] ) ) {
                $_desc = $term_to_ancestor[ $desc ];
                unset( $term_to_ancestor[ $desc ] );
                $desc = $_desc;
            } else {
                $desc = 0;
            }
        } while ( ! empty( $desc ) );
    }

    $possible_object_parents = array_filter( $possible_object_parents );

    $front_page_url = home_url();

    // if the menu item corresponds to the currently-queried post or taxonomy object
    if (
        $menu_item->object_id == $queried_object_id &&
        (
            ( ! empty( $home_page_id ) && 'post_type' == $menu_item->type && $wp_query->is_home && $home_page_id == $menu_item->object_id ) ||
            ( 'post_type' == $menu_item->type && $wp_query->is_singular ) ||
            ( 'taxonomy' == $menu_item->type && ( $wp_query->is_category || $wp_query->is_tag || $wp_query->is_tax ) )
        )
    ) {
        $classes[] = 'current-menu-item';
        // $menu_items[$key]->current = true;
        $_anc_id = (int) $menu_item->db_id;

        while(
            ( $_anc_id = get_post_meta( $_anc_id, '_menu_item_menu_item_parent', true ) ) &&
            ! in_array( $_anc_id, $active_ancestor_item_ids )
        ) {
            $active_ancestor_item_ids[] = $_anc_id;
        }

        $active_parent_item_ids[] = (int) $menu_item->menu_item_parent;
        $active_parent_object_ids[] = (int) $menu_item->post_parent;
        $active_object = $menu_item->object;

        // if the menu item corresponds to the currently-requested URL
        } elseif ( 'custom' == $menu_item->object ) {
            $current_url = untrailingslashit( ( is_ssl() ? 'https://' : 'http://' ) . $_SERVER['HTTP_HOST'] . $_SERVER['REQUEST_URI'] );
            $item_url = untrailingslashit( strpos( $menu_item->url, '#' ) ? substr( $menu_item->url, 0, strpos( $menu_item->url, '#' ) ) : $menu_item->url );
            $_indexless_current = untrailingslashit( preg_replace( '/index.php$/', '', $current_url ) );

            if ( in_array( $item_url, array( $current_url, $_indexless_current ) ) ) {
                $classes[] = 'current-menu-item';
                // $menu_items[$key]->current = true;
                $_anc_id = (int) $menu_item->db_id;

                while(
                    ( $_anc_id = get_post_meta( $_anc_id, '_menu_item_menu_item_parent', true ) ) &&
                    ! in_array( $_anc_id, $active_ancestor_item_ids )
                ) {
                    $active_ancestor_item_ids[] = $_anc_id;
                }

                if ( in_array( home_url(), array( untrailingslashit( $current_url ), untrailingslashit( $_indexless_current ) ) ) ) {
                    // Back compat for home link to match wp_page_menu()
                    $classes[] = 'current_page_item';
                }
                $active_parent_item_ids[] = (int) $menu_item->menu_item_parent;
                $active_parent_object_ids[] = (int) $menu_item->post_parent;
                $active_object = $menu_item->object;

            // give front page item current-menu-item class when extra query arguments involved
            } elseif ( $item_url == $front_page_url && is_front_page() ) {
                $classes[] = 'current-menu-item';
            }

            if ( untrailingslashit($item_url) == home_url() )
                $classes[] = 'menu-item-home';
        }

    // return the new array with classes
    return $classes;

}