no, this would not be done through the add to cart button.
it would be done through the shipping tab of the eCart object.
you could add a checkbox to the checkout page for next day shipping.
the checkout page submits to the confirm_cart page.
on the confirm_cart page, set a session based on the next day shipping selection, for example, if the checkbox is named "nextDay"
if(isset($_POST['nextday']) && $_POST['nextday'] != "") $_SESSION['nextDay'] = "1";
then in the eCart object, create a shipping rule using the trigger:
based on session variable value nextDay = 1
if you want to create multiple rules based on weight, add the weight conditions to the triggers as well.