I've been staying in the very center of Palermo for a week now, at a cool hostel called Palermo House. Good location, awesome vibe, friendly staff, hilariously small (but clean) bathrooms.

One of the better hostels I've ever stayed in, worldwide, in terms of atmosphere. Seems that I am going to stay a week or two longer than planned! Free continental breakfast is included. They also have a few private rooms too, either with shared bath or private bath.

Deets at http://palermohouse.com.ar/