functions.php

<?php
add_filter( 'manage_edit-shop_order_columns', 'woo_order_weight_column' );
function woo_order_weight_column( $columns ) {
  $columns['total_weight'] = __( 'Weight', 'woocommerce' );
	return $columns;
}

add_action( 'manage_shop_order_posts_custom_column', 'woo_custom_order_weight_column', 2 );
function woo_custom_order_weight_column( $column ) {
	global $post, $woocommerce, $the_order;

	if ( empty( $the_order ) || $the_order->id != $post->ID )
		$the_order = new WC_Order( $post->ID );

	if ( $column == 'total_weight' ) {
		$weight = 0;
		if ( sizeof( $the_order->get_items() ) > 0 ) {
			foreach( $the_order->get_items() as $item ) {
				if ( $item['product_id'] > 0 ) {
					$_product = $the_order->get_product_from_item( $item );
					if ( ! $_product->is_virtual() ) {
						$weight += $_product->get_weight() * $item['qty'];
					}
				}
			}
		}
		if ( $weight > 0 )
			print $weight . ' ' . esc_attr( get_option('woocommerce_weight_unit' ) );
		else print 'N/A';
	}
}
?>