After many hours of study and trying to figure this out I came up with a solution. I don't know if it is the best one but it worked! Here's the code that I added just after "foreach($attributes as $attribute) {" at the beginning in the file addtocart_advanced_attribute.tpl.php
$query = "SELECT * FROM `jos_vm_product` WHERE `product_id`='".$attribute['product_id']."'";
$rows = mysql_query($query) or die(mysql_error());
$product = mysql_fetch_array($rows);
This allowed me to call the Product Name in the attribute box as the variable "$product['product_name']".
I don't know if this will be useful to anyone else as it's not a common need but there it is just in case!