Hi,
There is actually no option to allow the user to set the cart name.
However, if your customers don't normally have several carts, I would recommend turning off the "Enable multi carts" setting.
That way, when you click on the "cart" button from the user control panel on the frontend, you'll directly see the current cart page, and not the listing of the carts.
And if you have create a menu item of the type "carts listing", I would recommend changing it to "current cart" so that the customer doesn't have to go through the listing of the carts and thus the "no name" text won't appear for him.