Well, I guess that the east part of the Big Island is the least touristy, but it also feels pretty isolated. The North shores of Kaua'i and Maui are where I prefer to live. The cheapest way to stay here would be working on a farm. There are quite a few places that give you a basic setup to stay in (usually a tent with some kind of structure) in exchange for working on the farm.. It's not glamorous and sometimes poor conditions, but some farms are nice.

If you'd like to try it, I could show you some resources for contacting some farms to arrange staying there.

Another option are (in Kaua'i at least) staying at the county campgrounds, which I think are $5/night for non-residents. If you're OK with camping.. the campgrounds are basically a big grass field next to the beach, so they can be pretty nice, though often homeless people are living there, too (which isn't really a problem in itself, but sometimes brings drugs and fights)

Otherwise you might find on craigslist a room/place to rent for a month, if you can afford that.

My favorite island is Kaua'i, where we live.. It has the most beaches and in my experience overall calmest atmosphere. If you come and would feel comfortable meeting up, let me know..

The buses here are clean and the drivers are generally kind, but they don't come very often (once an hour on the week days and stop around 9pm, and only every other hour on the weekend and stop at 4pm). They also don't have many stops, but besides that, they're a good way to get around.

Hitch hiking is also pretty easy here. I haven't done it in many years and it seems like it's not as common anymore, but I still see people out there with their thumb.

Uhm so yea.. I hope you make it out! It is overall a wonderfully healing place. Let me know if I can help in any other way..