I am trying to identify the lacing material on these sandals. The photo is of the original Luna Huarache sandal. They call it an "elasticized leather".
The manufacturer description can be interpreted quite simply. "Elasticized leather" is not some special sort of leather - that phrase is just saying that the straps are made of leather which has had elastic attached to it.
If you do a search for "elasticized leather" you will find this phrase used elsewhere, for example an "elasticized leather waistband" on a jacket, or "elasticized leather panel" on gloves. You can see that this just refers to sections of the article which have had elastic applied.
Looking at the image, there appears to be a zig-zag stitch along the straps. This is how the elastic is held in place here.
To recreate these straps, you would sandwich a elastic between suitably flexible leather. The elastic should be shorter than the leather (otherwise it wouldn't have any effect). Sew through the layers with a zig-zag stitch centered on the elastic, stretching the elastic as you go to match the length of the leather. When you're done, the elastic will snap back to size, resulting in a stretchy ("elasticized") leather strap.