How to display the title for each menu item in a span attribute

When calling the wp_nav_menu() function you can pass the <span> tags in the link_before y link_after arguments:

wp_nav_menu(array('link_before' = > '<span>', 'link_after' => '</span>'));

To have differents attributes in the span for each menu item you may need to use a custom nav_walker. For example, the next custom Nav_Walker is the default from WordPress, I’ve just modified it to add the <span> with the data-hover attribute:

//First, define the callback when displaying the menu
<nav>
    <?php wp_nav_menu( array( 
            'container'       => '',
            'container_class' => '',
                'theme_location'  => 'topmenu',
            'menu_class'      => 'nav',
                'fallback_cb'      => '',
            'walker' => new Custom_Walker(),
                     ) );
?>

The custom nav walker, in this case Custom_Walker must be in functions.php or included in it:

class Custom_Walker extends Walker_Nav_Menu {
/**
 * @see Walker::$tree_type
 * @since 3.0.0
 * @var string
 */
var $tree_type = array( 'post_type', 'taxonomy', 'custom' );

/**
 * @see Walker::$db_fields
 * @since 3.0.0
 * @todo Decouple this.
 * @var array
 */
var $db_fields = array( 'parent' => 'menu_item_parent', 'id' => 'db_id' );

/**
 * @see Walker::start_lvl()
 * @since 3.0.0
 *
 * @param string $output Passed by reference. Used to append additional content.
 * @param int $depth Depth of page. Used for padding.
 */
function start_lvl( &$output, $depth = 0, $args = array() ) {
    $indent = str_repeat("\t", $depth);
    $output .= "\n$indent<ul class=\"sub-menu\">\n";
}

/**
 * @see Walker::end_lvl()
 * @since 3.0.0
 *
 * @param string $output Passed by reference. Used to append additional content.
 * @param int $depth Depth of page. Used for padding.
 */
function end_lvl( &$output, $depth = 0, $args = array() ) {
    $indent = str_repeat("\t", $depth);
    $output .= "$indent</ul>\n";
}

/**
 * @see Walker::start_el()
 * @since 3.0.0
 *
 * @param string $output Passed by reference. Used to append additional content.
 * @param object $item Menu item data object.
 * @param int $depth Depth of menu item. Used for padding.
 * @param int $current_page Menu item ID.
 * @param object $args
 */
function start_el( &$output, $item, $depth = 0, $args = array(), $id = 0 ) {
    $indent = ( $depth ) ? str_repeat( "\t", $depth ) : '';

    $class_names = $value="";

    $classes = empty( $item->classes ) ? array() : (array) $item->classes;
    $classes[] = 'menu-item-' . $item->ID;

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

    $id = apply_filters( 'nav_menu_item_id', 'menu-item-'. $item->ID, $item, $args );
    $id = $id ? ' id="' . esc_attr( $id ) . '"' : '';

    $output .= $indent . '<li' . $id . $value . $class_names .'>';

    $atts = array();
    $atts['title']  = ! empty( $item->attr_title ) ? $item->attr_title : '';
    $atts['target'] = ! empty( $item->target )     ? $item->target     : '';
    $atts['rel']    = ! empty( $item->xfn )        ? $item->xfn        : '';
    $atts['href']   = ! empty( $item->url )        ? $item->url        : '';

    $atts = apply_filters( 'nav_menu_link_attributes', $atts, $item, $args );

    $attributes="";
    foreach ( $atts as $attr => $value ) {
        if ( ! empty( $value ) ) {
            $value = ( 'href' === $attr ) ? esc_url( $value ) : esc_attr( $value );
            $attributes .= ' ' . $attr . '="' . $value . '"';
        }
    }

    $item_output  = $args->before;
    $item_output .= '<a'. $attributes .'>';
    $item_output .= $args->link_before;
            // HERE are your <span> tags
            $item_output .= '<span data-hover="'.esc_attr($item->title).'">';
            $item_output .= apply_filters( 'the_title', $item->title, $item->ID );
            $item_output .= '</span>';
            $item_output .= $args->link_after;
    $item_output .= '</a>';
    $item_output .= $args->after;

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

/**
 * @see Walker::end_el()
 * @since 3.0.0
 *
 * @param string $output Passed by reference. Used to append additional content.
 * @param object $item Page data object. Not used.
 * @param int $depth Depth of page. Not Used.
 */
function end_el( &$output, $item, $depth = 0, $args = array() ) {
    $output .= "</li>\n";
}
}

Or maybe better (or not, I’m not sure because it needs a query foreach menu item via the get_post() function if we want to run the filter only for the title of menus and not to all post titles), use the_title filter hook (see toscho answer for a better way to use the_title filter hook if you decide to use this filter instead the custom nav-walker):

function add_span_to_menu($title, $id) {

    //Run this function only for nav_menu_item post types
    $item = get_post( $id );
    if($item->post_type != 'nav_menu_item') return;

    $title="<span data-hover="".esc_attr($title).'">'.$title.'</span>';

    return $title;

}
add_filter('the_title', 'add_span_to_menu', 10, 2);