Their northern cousins wore roll hats that were about the thickness of a forearm at most. Well, anything you'd wear to an SCA event. In general, use woven natural fabrics, such as cotton, linen or wool. If you're not sure what sort of hat you want, or don't trust your design skills, copy one of the ones in this handout, or from a medieval illustration. Cut a piece of fabric 1 yard long by whatever diameter roll you want.
Some folks swear by cutting on the bias and stiff stuffing to eliminate interior wrikles.
Stuff with stuffing. Try a tall thin drinking glass or other cylindrical object. The idea with the stuffing is to get it firmly enough to keep the shape, but not so dense that it can't compress very well. You'll have to experiment. A friend recalls seeing a round cork shape in a museum which was made to be rolled up inside a hood to give the stuffed effect, but I do not have any source material on this option.