This well-run hostel in a quiet courtyard is a great option. It's centrally located on Trinity Hill, and has 40 beds in four- to eight-bed dorms (some female) and several excellent-value private rooms distributed between the main building and an adjoining riverside annex. There's a no-alcohol rule and a strict ban on making noise after 11pm.
This is not a place to come to party. To find it, walk north along the riverside walk of Trinity Hill and turn into the last arch before the road ends.