I was searching for a solution but didn't find any online. But eventually found a workable solution in one of the email templates that already has the customer name.
It is in New Order email for admin (admin-new-order.php). It has a nice snippet which says "You have received an order from %s"
So, just copy that line of code to whichever email template you want to use it and of course change the text to something else
<p><?php printf( __( 'You have received an order from %s. Their order is as follows:', 'woocommerce' ), $order->billing_first_name . ' ' . $order->billing_last_name ); ?></p>
Make sure you copy the email template from
before you edit so that new Woocommerce version won't overwrite your changes.