Straw adds tensile strength to the cob.
If you have a clay-rich soil and limited sand, you can use a heavier clay proportion by adding extra straw. If you have a very sandy mix, it won't be able to hold as much straw. That's OK.
Straw is the stalks of grasses or grains left after the nutritious seed heads and leaves have been removed. It's cheap stuff, often baled for easy handling and transportation. The fresher it is, the stronger it is, so avoid old straw that's started to decompose. Make sure to keep the straw dry until you're ready to use it. Do not use hay. It decomposes.
Some people think certain lengths are better than others but I just throw it in as it comes out of the bale and it seems to work great. The straw will soak up some of the water in your cob mix, drying it slightly.
Get at least 10 bales for making a small cottage or for each large room. More is good.
Extra straw bales are wonderful to have at the building site. They serve as scaffolding supports, ladders, walls for clay-making pits, chairs, backrests, tables, even beds. Eventually they'll become great garden mulch or erosion control.
If you're an ambitious purist, you can gather your own grass stalks or experiment with other plant fibers.
I usually use plain fresh water in the cob mixes. I have tried all of the following ideas and they all seem to work fine.
When I first started making cob, I used to turn it with a garden fork. Mixing the cob on a tarp is a much easier way to stir the ingredients and save your back, so get a good supply of tarps. Any big tarp of sturdy material will do (7x9 foot or bigger). Those awful woven blue plastic ones work well. They're light and the dirt doesn't stick to them much, but they do disintegrate after a while. The fibers from the shredded tarps can be added to the cob mix for reinforcing and to keep them out of the landfills, tips or dumps. Old awnings from recreational vehicles are good for the job too, and last longer than the tarps. RV repair shops are often happy to get rid them.