Cork
Ok, so Cork isn't really a fabric, but I wanted to add it to this section.
Cork you buy backed with Organic Cotton which makes it a very strong and abrasion resistant material. Please make sure if you buy it that it isn't not backed with polyester! The cork is taken from the Cork Oak tree which is mature enough to harvest after 9 years, the same tree can then be harvested for the next 200 years!
You can also buy Cork cord, which is a really decent material. Unfortunately this is usually cork wrapped around a plastic or Polyurethane cord, so not particularly great for the environment. Even more unfortunately you will find people shouting about it's eco friendly credentials. Try to find the Organic Cotton plaited cork cord... it's not easy to find, but does exist!