Whatever fabric you choose, it should be something woven rather than knit, non-stretchy, and not too slippery/silky.
One of my favorite fabrics is linen or a linen blend. I find that the looser weave of a linen fabric (as opposed to something like a dense cotton twill) gives just a little bit of stretch and makes the sling very comfortable.
Another really nice fabric, if you want something a bit more exotic looking, is silk
dupioni, which has a nubby texture. Silk is lightweight, strong, and very durable. I especially like a two-tone
dupioni woven with different colors of warp and weft, which makes it appear to change colors as you move. A
dupioni's weave gives it enough "grip" so that it doesn't slip out of the rings. And even though silk has a reputation as a delicate fabric, it's actually machine washable.
You will see the back of the fabric on the tail, so select something that looks nice on both sides.
I will need 2 1/4 yards for a medium length sling, and 2 1/2 yards for a long one. Most people fit a medium length sling, but if, for example, you're married to a linebacker-sized guy, you'd probably want a longer sling.
I also need a spool of matching thread.