This is how you might add the product categories to the Product title. I don't particularly like how it is outputting for me, but then I am testing it on a variable subscription, so that is amending its own data. This would go in your theme's functions.php
function kia_woocommerce_order_item_name( $name, $item ){
    $product_id = $item['product_id'];
    $tax = 'product_cat';
    $terms = wp_get_post_terms( $product_id, $tax, array( 'fields' => 'names' ) );
    if( $terms && ! is_wp_error( $terms )) {
        $taxonomy = get_taxonomy($tax);
        $name .= '<label>' . $taxonomy->label . ': </label>' . implode( ', ', $terms );
    }
    return $name;
}
add_filter( 'woocommerce_order_item_name', 'kia_woocommerce_order_item_name', 10, 2 );
Otherwise you could copy the order-details.php template into your own theme (so yourtheme/woocommerce/order/order-details.php and add the code directly. This would give you more control over where you want the category to appear since there aren't a lot of hooks available in this section, but put you at risk if WooCommerce ever modifies this code. I have worked on a few sites that broke because the the theme was using outdated template. 
Any way, an example would be:
<?php
/**
 * Order details
 *
 * @author      WooThemes
 * @package     WooCommerce/Templates
 * @version     2.2.0
 */
if ( ! defined( 'ABSPATH' ) ) exit; // Exit if accessed directly
global $woocommerce;
$order = new WC_Order( $order_id );
?>
<h2><?php _e( 'Order Details', 'woocommerce' ); ?></h2>
<table class="shop_table order_details">
    <thead>
        <tr>
            <th class="product-name"><?php _e( 'Product', 'woocommerce' ); ?></th>
            <th class="product-total"><?php _e( 'Total', 'woocommerce' ); ?></th>
        </tr>
    </thead>
    <tfoot>
    <?php
        if ( $totals = $order->get_order_item_totals() ) foreach ( $totals as $total ) :
            ?>
            <tr>
                <th scope="row"><?php echo $total['label']; ?></th>
                <td><?php echo $total['value']; ?></td>
            </tr>
            <?php
        endforeach;
    ?>
    </tfoot>
    <tbody>
        <?php
        if ( sizeof( $order->get_items() ) > 0 ) {
            foreach( $order->get_items() as $item ) {
                $_product     = apply_filters( 'woocommerce_order_item_product', $order->get_product_from_item( $item ), $item );
                $item_meta    = new WC_Order_Item_Meta( $item['item_meta'], $_product );
                ?>
                <tr class="<?php echo esc_attr( apply_filters( 'woocommerce_order_item_class', 'order_item', $item, $order ) ); ?>">
                    <td class="product-name">
                        <?php
                            if ( $_product && ! $_product->is_visible() )
                                echo apply_filters( 'woocommerce_order_item_name', $item['name'], $item );
                            else
                                echo apply_filters( 'woocommerce_order_item_name', sprintf( '<a href="%s">%s</a>', get_permalink( $item['product_id'] ), $item['name'] ), $item );
                            echo apply_filters( 'woocommerce_order_item_quantity_html', ' <strong class="product-quantity">' . sprintf( '× %s', $item['qty'] ) . '</strong>', $item );
                            $item_meta->display();
                            // insert product category here
                            $tax = 'product_cat';
                            $terms = wp_get_post_terms( $_product->id, $tax, array( 'fields' => 'names' ) );
                            if( $terms && ! is_wp_error( $terms )) {
                                $taxonomy = get_taxonomy($tax);
                                echo '<strong>' . $taxonomy->label . ': </strong>' . implode( ', ', $terms );
                            }
                            // end edit
                            if ( $_product && $_product->exists() && $_product->is_downloadable() && $order->is_download_permitted() ) {
                                $download_files = $order->get_item_downloads( $item );
                                $i              = 0;
                                $links          = array();
                                foreach ( $download_files as $download_id => $file ) {
                                    $i++;
                                    $links[] = '<small><a href="' . esc_url( $file['download_url'] ) . '">' . sprintf( __( 'Download file%s', 'woocommerce' ), ( count( $download_files ) > 1 ? ' ' . $i . ': ' : ': ' ) ) . esc_html( $file['name'] ) . '</a></small>';
                                }
                                echo '<br/>' . implode( '<br/>', $links );
                            }
                        ?>
                    </td>
                    <td class="product-total">
                        <?php echo $order->get_formatted_line_subtotal( $item ); ?>
                    </td>
                </tr>
                <?php
                if ( $order->has_status( array( 'completed', 'processing' ) ) && ( $purchase_note = get_post_meta( $_product->id, '_purchase_note', true ) ) ) {
                    ?>
                    <tr class="product-purchase-note">
                        <td colspan="3"><?php echo wpautop( do_shortcode( $purchase_note ) ); ?></td>
                    </tr>
                    <?php
                }
            }
        }
        do_action( 'woocommerce_order_items_table', $order );
        ?>
    </tbody>
</table>
<?php do_action( 'woocommerce_order_details_after_order_table', $order ); ?>
<header>
    <h2><?php _e( 'Customer details', 'woocommerce' ); ?></h2>
</header>
<dl class="customer_details">
<?php
    if ( $order->billing_email ) echo '<dt>' . __( 'Email:', 'woocommerce' ) . '</dt><dd>' . $order->billing_email . '</dd>';
    if ( $order->billing_phone ) echo '<dt>' . __( 'Telephone:', 'woocommerce' ) . '</dt><dd>' . $order->billing_phone . '</dd>';
    // Additional customer details hook
    do_action( 'woocommerce_order_details_after_customer_details', $order );
?>
</dl>
<?php if ( ! wc_ship_to_billing_address_only() && $order->needs_shipping_address() && get_option( 'woocommerce_calc_shipping' ) !== 'no' ) : ?>
<div class="col2-set addresses">
    <div class="col-1">
<?php endif; ?>
        <header class="title">
            <h3><?php _e( 'Billing Address', 'woocommerce' ); ?></h3>
        </header>
        <address><p>
            <?php
                if ( ! $order->get_formatted_billing_address() ) _e( 'N/A', 'woocommerce' ); else echo $order->get_formatted_billing_address();
            ?>
        </p></address>
<?php if ( ! wc_ship_to_billing_address_only() && $order->needs_shipping_address() && get_option( 'woocommerce_calc_shipping' ) !== 'no' ) : ?>
    </div><!-- /.col-1 -->
    <div class="col-2">
        <header class="title">
            <h3><?php _e( 'Shipping Address', 'woocommerce' ); ?></h3>
        </header>
        <address><p>
            <?php
                if ( ! $order->get_formatted_shipping_address() ) _e( 'N/A', 'woocommerce' ); else echo $order->get_formatted_shipping_address();
            ?>
        </p></address>
    </div><!-- /.col-2 -->
</div><!-- /.col2-set -->
<?php endif; ?>
<div class="clear"></div>
 
				
woocommerce_order_item_namefilter. - helgathevikingwoocommerce_order_item_name. I'm going to go check it out now. Please let me know if you think I should add more to my question - JohnMcCarthy