Hop on the website Booking.com to start and look in each town for whatever type of lodging meets your budget and is also in a decent location. (In some of the towns, your lodgings could be up some steep steps, FYI!)
I've stayed in the Cinque Terre twice as a solo traveler. Both times, I wound up staying in Levanto, the village to the north of Monterosso, just one train stop north. First time, I stayed at a hotel; last time, i stayed at a B&B-type place. Both were fine; the B&B was a lot cheaper. Levanto is itself a nice town - more of a conventional "beach town" than the "build into the side of the hill' feel of the main five villages, but I liked Levanto and would stay again. If you stay there, choose a location within 10-15 minutes walk of the train station (in the town, a few blocks from the beach if possible). it's easy to get between all towns by train - very frequent train service.
You could even stay further away than Levanto if you wish - e.g. in Sestri Levante, if by chance you can find something at a great price there but day trip by train to the Cinque Terre. I'd choose Levanto if prices are close at all, though.