Like may things in life, the deeper you explore the Big Island, the more treasure it reveals.
Hiking: Hike to deserted beaches, waterfalls, sea arches, ancient Hawaiian Petroglyphs, green sands, volcanic caves and craters, and the list goes on and on. The Big Island's variety of terrains, geological peculiarities, and stunning vistas make it a hiker's paradise. Many of Big Island's most worthy sights are for the hiker only. Some popular hiking areas are Volcano's National Park (see Volcano), Kiholo Bay, Waipi'o Valley, and Wai'ale Falls. For specific trails check out some of the many hiking guides available at Borders Bookstore in Kailua-Kona or check out some of the listed websites.
Cave Exploration: Since the Big Island is such a young island, lava tubes are more common here than anywhere else in the state. Along with the Thurston Lava Tube in Volcano National Park, you can explore other lava tubes by taking various cave tours. Make sure to bring your shoes!
Skiing: Really? Skiing in Hawaii? Yup! Or how about bodyboard sledding? At just under 13,800 ft high, Mauna Kea often gets lots of snow in the winter. Conditions vary and 4 wheel drive is needed (both as a means to get there and to act as your ski lift), but if you want to say you skied in Hawaii here's your chance. Ski Guides Hawaii rents gear and gives you the scoop.
Stargazing: Thirteen telescopes operated by astronomers from multiple countries take advantage of Mauna Kea summit's pristine air conditions, making it the largest astronomical observatory in the world. Well above the tropical cloud layer, see the heavens like you have never seen them before at the Visitor's Center of Onizuka's Center for International Astronomy (9,000 ft) or take a guided tour to the summit (14,000ft). Tours commonly include transportation to Mauna Kea, warm clothes, and dinner.