If you are vacationing, then booking a hotel a few blocks back from the beach will cut costs. Look for a place that is on Kuhio Ave or on one of the side streets that run between Kalakaua the Ala Wai. As far as living in Hawaii, the Big Island is less expensive than most places. On Oahu the further you get from Honolulu,the less expensive it is. But that would include neighborhoods that would not be very hospitable to transplants form the mainland.
Cheapest place would be the neighborhoods that the resort staff making minimum wage are living in.
nowhere! everywhere is pricey there!
In a dumpster.