To add product weight on WooCommerce archive pages, right below the product title, add the following code to your child theme functions.php file or to a plugin such as Code Snippets that allows adding functions like this.
WooCommerce 3.0+
/**
* Show product weight on archive pages
*/
add_action( 'woocommerce_after_shop_loop_item', 'savvy_show_weights', 9 );
function savvy_show_weights() {
global $product;
$weight = $product->get_weight();
if ( $product->has_weight() ) {
echo '<div class="product-meta"><span class="product-meta-label">Weight: </span>' . $weight . get_option('woocommerce_weight_unit') . '</div></br>';
}
}