There is currently only one hostel in the Hilo area: Howzit Hostel. If you are looking for other affordable options to spend the night, please refer to our camping on the Big Island resource.
Howzit Hostel in Hilo
Howzit Hostel is a clean, modern hostel in downtown Hilo, offering a convenient base near shops and restaurants, with dorm beds starting around $60 per night including taxes.
Address: 107 Waianuenue Avenue, Hilo, Hawaii 96720
Website: https://www.howzithostels.com/hilo

The hostel includes a lounge, shared kitchen, and evening entertainment. Additional facilities include a 24-hour front desk, housekeeping service, full-day security, and free parking. Image: Howzit Hostel website.
Affordable accommodation in Puna
Puna has a fair share of affordable (shared) accommodation options. Many of them pop in an out of existence too often for us to recommend any but the Open Gate Hostel on the lava fields in Kalapana seems to have found their stride.
You can find a good sample of the current offerings at e.g. the Hipcamp website.