Select a rope that is at least 3/4-inch in diameter. You'll have a lot of choices from that point on, but if you're looking for a natural fiber look, something that is resistant to sunlight, heat and abrasion (you're wanting it to last, right?), then manila is your choice.
Check the label to be sure, but generally manila rope has a working load limit of 695 pounds. You can certainly find higher load limits with twisted polypropylene (1,090 pounds and about 15% more per foot), or twisted nylon and polyester (1,420 pounds about 80% more per foot), but you'll spend a good bit more and not have quite the resistance features as manila in the elements. Get plenty of rope — 15 to 20 feet — because you certainly don't want to be short any rope. A good rule of thumb is to set the bottom of the tire at least 12 to 18 inches from the ground.
If cost isn't an issue, you can simply buy a single new tire. But realistically, a used tire is what you're after. There are used tire shops that will sell you a single tire. You can also try chain tire stores to see if you can buy one. This will depend on the chain's policy. You can try junkyards or pull-a-part type places as well. And luck may have it that you spot a tire at a garage sale or flea market.
Make sure the tire is in decent shape without large fissures and cracks. Also look out for thin tread where the reinforcing wires may be revealed, a definite issue to avoid.