For our sling, we found that a rectangle of denim, with the inside coated with rubber cement, worked better than any other design. A cotter pin put through three fencing staples worked well. That way, we were able to attach a cord to the cotter pin and pull it out from a distance. If you're looking for historical accuracy, go with a free-hanging weight. If you're looking for distance, use bungees or rubber snubbers.
Bungees work, but because of the way they're made they have a lot less stretch before they break.
Rubber Snubbers should be of the heavy-duty, extra stretchability variety. The track the sling slides through on its way out and around should be able to guide the sling, but should be no narrower than the sling. Friction doesn't really matter, as it's effect on the entire system is negligible.