The Boardwalk Bakery is the closest place to refill your mug if you are at the BWI. Hours vary by season. When we were there in February, the hours were 7 am - 11 pm. You can also get a free refill at Boardwalk Joe's Coffee Cart. The hours (in February) were 8 am - 10 pm).
It's been reported elsewhere on these boards that you can also get the free refills at the pool - at the Leaping Horse Libations Pool Bar. Hours in February were 11 am to dusk.