Instructions:

1. Copy/paste this into your functions.php (TODO: convert it to a standalone php file, and make it into a plygin you can activte/deactivate)
2. Create a new menu item of type “custom URL”
3. Make your URL “http://#latestpost:category_name”
   • where “category_name” is the name of the category whose latest post you want to link to
4. Make the name whatver you want to appear on the menu
5. Profit!

Based on an idea (with some upgrading + bugfixes for latest WordPress in 2016) from http://www.viper007bond.com/2011/09/20/code-snippet-add-a-link-to-latest-post-to-wordpress-nav-menu/

/** Adam: add support for putting 'latest post in category X' to menu: */
// Front end only, don't hack on the settings page
if ( ! is_admin() ) {
    // Hook in early to modify the menu
    // This is before the CSS "selected" classes are calculated
    add_filter( 'wp_get_nav_menu_items', 'replace_placeholder_nav_menu_item_with_latest_post', 10, 3 );
}

// Replaces a custom URL placeholder with the URL to the latest post
function replace_placeholder_nav_menu_item_with_latest_post( $items, $menu, $args ) {

        $key = 'http://#latestpost:';

    // Loop through the menu items looking for placeholder(s)
    foreach ( $items as $item ) {
 
        // Is this the placeholder we're looking for?
        if ( 0 === strpos( $item->url, $key ) )
        {
 
        $catname = substr( $item->url, strlen($key) );
        // Get the latest post
        $latestpost = get_posts( array(
            'posts_per_page' => 1,
                'category_name' => $catname
        ) );

        if ( empty( $latestpost ) )
            continue;

        // Replace the placeholder with the real URL
        $item->url = get_permalink( $latestpost[0]->ID );
        }
    }

    // Return the modified (or maybe unmodified) menu items array
    return $items;
}