There actually is a way to do this in Manager, but it is not as easy as checking a checkbox, it involves making a custom product template.
Basically, the only way the shopping cart knows to add the parent product's name to the subproduct name is that the subproduct link (or form) has an extra parameter in it (super=[itemnum_of_parent]) that indicates the item number of the parent product. If that were left off, even though the product being added is assigned as a subproduct the cart wouldn't display it with the parent product name because it wouldn't even know that there is a parent product. But all default product template tags that are normally used to create an add to cart button for a subproduct automatically add this value to the add to cart link (or form).
But you could make a custom product template and in the DEFINE SUBPRODUCT section don't use the normal template tags that create the add to cart form or link, instead manually create the form or link from more atomic template tags. For example, instead of [-- PRODUCT.OrderCheckout --] you can use:
<a href="[-- SHOPPING_CART_URL BASE --]/order.cgi?storeid=[-- STORE.ID --]&dbname=products&itemnum=[-- PRODUCT.RecordNumber --]&function=add">Add to Cart</a>
Which will create an Add to Cart link for the product wihout the parent products name associated with it. If you also send Quantity and/or Ordering Options to the cart with the subproducts, then it gets a bit more complicated because you have to make it as a form instead of just an href link, but a good example of this is in the documentation at:
http://www.shopsite.com/templates/examp ... tions.html
-Loren