It depends where on the Big Island you're going to be. The Big Island is big.
My favorite things to do on the Hilo side:
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Uncle Robert's Awa Bar and outdoor Farmers Market on Wednesday evening in Kalapana (go early and take a pre-sunset walk on the lava field that used to be the bay of a black sand beach; if you walk out far enough, you'll find the coconut grove started by Uncle Robert's wife, and a new black sand beach forming at the ocean.
>> Pahoa. The town just north of Kalapana. Shops and restaurants
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Ahalanui hot pond in the Puna District, just to the east of Kalapana.
>> Hilo Guitar & Ukulele. Great shop.
>> Helicopter ride over the volcano from Hilo Airport
>> Volcanoes National Park. You can hike the old and now dormant crater, walk through the Thurston Lava Tunnel, and see the volcano erupting.
Mauna Kea
>> You can drive up to the visitor center and hike from there to the summit to watch the sun set. Afterwards, there;'s usuqally an astronomy talk and a telescope you can use at the visitor center. If you don't want to hike to the summit and back (it's not a long hike, but you'd be coming back in the dark), there are small bus sunset tours.
On the Kona side, outside of Kona:
>> Snorkeling at Captain Cook. You can go on a snorkeling boat tour, or you can hike to the Captain Cook Monument from the town of Captain Cook
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Hawi. It's a small town north of Kona, on the northwest tip of the island. There's some Kamehameha stuff in the area and a cool shop right in Hawi that has some vintage 'ukuleles and other Hawaii stuff.
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The Parker Ranch in Waimea. North of Kona, on the way to Hawi.
Have fun! Take and post photos!