Located just to the East of Greymouth is Arthur's Pass National Park, which is a major hub for hiking and adventure. Although the entire town of Greymouth would be considered "off the beaten path" in my opinion, this is one of the places I would go if in Greymouth for an extended period of time.
For more information on Arthur's Pass, I encourage you to take a look at my page:
During the planning for a trip to Greymouth, you will likely be using this as a stop on your way around the West Coast of the South Island. Your final destination will likely be Hokitika or the Glaciers, but I encourage you to take a slightly northern path to Punakaiki from Greymouth. It is about 45km north, but worth the extra two or three hours on your way.
The Paparoa National Park is also located just north of Punakaiki, and would make for a lovely day trip if you are staying in Greymouth.
For more information on this trip, check out this page: