How do we make a specific shipping option free or discounted?

One way is to use a shipping tag as the outcome on the discount rule in Brink, then configure that tag with your shipping provider so it resolves to the price you intend. The tag travels to the provider the same way the standard free-shipping tag does. This is the mechanism for both free shipping on campaigns and reduced shipping for specific customer groups.

It is also possible to use the shipping providers own framework for free shipping. Please reach out to your shipping provider to get more information.