Thanks to my friends, Frank and Kristin Hoffman at http://www.knot4fun.com/, I rebuilt our rope swing over the weekend. I used a swing hitch to secure the rope to a Sweet Gum branch which overhangs our back yard.
I used solid braid, multi-fiber (polypropylene, nylon, polyester) rope, 7/16" diameter (purchased at Home Depot). This has a working load of 300lbs, so I doubled up the rope. I tied a swing hitch at the top, and secured the loose end with a stick so it cannot work out of the knot. I then used overhand knots on the rope all the way down so that the kids can climb up the rope.
At the seat, I use a 2x4 with a single hole drilled through the middle. Thread the rope through the hole & tie an overhand knot under the seat bottom so the 2x4 rests on that knot. I leave a little 'tail' on the rope that I can use as a handle when giving the kids a big push on the swing.
I chose my rope so that it won't stretch with age or use, nor will it shrink after a rain. The swing is about 13 feet high off the ground.