There's only one hostel that I'm aware of - it's in the third town of the Cinque Terre at Corniglia.
I would recommend staying at Riamaggiore - it's the first town of the Cinque Terra, has a number of restaurants and accommodation options, but still has plenty of character. It also means that you can walk the Cinque Terre from Riomaggiore to Monterosso (far easier doing it that way than in reverse), and then catch the train back.
Both times I've been to the Cinque Terre I've just arrived at Riamaggiore and visited one of the accommodation provider officers in the main street. On each occasion I received excellent apartments (the first time, a three bedrooom apartment for 60 euro, and the second time a one bedroom apartment for 30 euro) - both were spotlessly clean and centrally located. over 10 years ago
In April 2011, we stayed in Vernazza with Giuliano. Here is some info...Giuliano Basso, ph: +39 333 341 4792 rents four lovely, well kept rooms with a view of the ocean and the village. Giuliano is a rare Vernazza native and a gracious host. He and his (American) wife Michele maintain these lovely rooms proudly built by Giuliano. From €70 a night. about 8 years ago