I :heart: :heart: :heart: the Big Island, and would love to ramble around on a bike there one day.
The one thing I thought of was Saddle Road. It connects Kona and Hilo, and it used to be a scary, only partially paved and partially maintained road and was even banned on car rental agreements. However, there has been quite a bit of maintenance done on it lately, no more bans (I think), so you might check it out. Pretty cool area, turns into a kind of a no-mans land through the middle of the island.
I would drive it first to be sure nothing has changed (road washed out, etc.) - there is a whole lot of nothing out there. Also you will want lots of water, nutrition, etc. There aren't 7-11s nearby. But, man, it seems cool.
Riding around the exterior roads would be fun too. A lot wetter, for sure.
Maybe check out the Ironman routes?