I live in Honolulu and the Big Island is the neighbor island I've visited the most. On at least 2 occasions, I've stayed there an entire week, and activities were never lacking. We usually stay at the Hilton Waikoloa on the Kohala Coast, which is an attraction in itself!
Like DawnCt1 said, the Big Island truly is big! Keep in mind that it will take at least 2 1/2 to 3 hours to get to Hilo on the other side of the island, but it's doable in a day trip.
I agree that you should check out the Volcano, which is kind of on the other side of the island. You can drive around the southern side of the island to Kau, which is the southern-most tip of the United States. There's not much there, but the sheer cliffs are pretty amazing.
In Hilo, be sure to stop off at Big Island Candies and pick up their chocolate covered shortbread cookies. Their chocolate covered macadamia nuts are very good too. They make great gifts. They only sell them on the Big Island, or online thru their website. North of Hilo is the Hamakua Coast, which is breathtaking. It rains a lot there, so it's very green, and there are a lot of cliffs, making for great views. Lots of waterfalls. Be sure to go to Akaka Falls along that coast.
There is also a road between Kohala and Hilo called the Saddle Road. There are ranches and farmlands there. In Kamuela, along Saddle Road there are a couple of well-known restaurants: Edelweiss and Merrimans. I haven't tried Edelweiss yet, but we stopped at Merrimans for lunch and really enjoyed it. If you get a chance to try Kamuela beef (I think quite a few Big Island restaurants serve it), try it, it's excellent!
In Kawaihae, just north of Kahala, there's a restaurant called Cafe Pesto that we really enjoy. I think they have a couple of other locations too, but that's the only one I've been to. Also in Kawaihae, there's a National Monument (the name escapes me right now -- I think it was a burial ground).
The northern part of the island, Hawi, is beautiful too. We drove around there and came by a pasture with wild horses. There is also a hiking trail there. The King Kamehameha statue is there, and across the street is a little hole in the wall counter service place, I think it was called Rainbow Cafe, that was great. I tried a lamb burger, and it was one of the best burgers I've ever had! There is also a flume thingy there -- again, the name escapes me, but you ride in these banana-looking rafts down a flume.
One year we did a Zodiac raft ride that left from Honokohau Harbor (south of Kona) and went to Kealakekua Bay, where Captain Cook arrived. It was really one of the best things I've ever done on vacation, we had dolphins swimming really close to our raft, and we were able to go into some lava tubes/caves.
Anyhow, I'm sure there are more things I'll think of later, but I'm almost certain you can find nice ways to fill 7 days there! Also, I'm not a golfer, but I don't think the island is lacking in that area either!
