How to add customer name to new customer admin order email

It would be tempting to use: {customer_name}. Sadly this is not supported by the core WooCommerce plugin, but it is available if you use the WooCommerce Follow-ups extension:
https://docs.woocommerce.com/document/automated-follow-up-emails-docs/email-variables-and-merge-tags/

Alternatively you could copy the email template to:
themes/your-child-theme-name/woocommerce/emails/…

The $order object is passed to the template, so adding the following code to the template should be able to recover the customer’s name from the order:

  $user_id = $order->get_customer_id();
  if( !$user_id ) {
    return; // user_id not found
  }
  $user = get_user_by( 'ID', $user_id );
  if( !$user ) {
    return; // user not found
  }
  $customer_name = $user->first_name.' '.$user->last_name;
    
  // email header
  do_action( 'woocommerce_email_header', $email_heading, $email );
?>

// will have to go in the message body - modifying the subject is too complex for here.

<p>Customer name: <?php print $customer_name; ?></p>

Some php skills will be needed to get it working.

So I can’t add any of these to the subject line then? The only way is to buy yet another extension or create a new php page?

{customer_username} Displays the username of your customer
{customer_first_name} Displays the first name of your customer
{customer_last_name} Displays the last name of your customer
{customer_name} Displays the concatenated first and last name of your customer

Not with the core plugin.

Regarding the Woo extension, and I can’t say whether that will allow subject line placeholders as well as message body placeholders. They will take pre-sales questions.

I found this free plugin:
https://www.ads-software.com/plugins/woo-custom-emails/
It has an {order_billing_name} placeholder, amongst others, but it looks like that will only work in the message body and not in the subject line.

Check out other email plugins at www.ads-software.com/plugins and codecanyon.net. There are a few. Its unlikely anyone will know all of them and you’ll need to just try a few promising ones.

With php skills or a developer it will be achievable. You can post a job here:
https://jobs.wordpress.net/