Rope Material in Colonial America

I'm thinking that hemp and flax (linen) would be the two common, cheap materials, as they could be locally grown. When cotton production started up, they could use cotton cord as well.

There are a lot of other fibers, but most are imported from tropical countries where they grow. I don't know when they would be available, let alone the comparative costs.

There's not much difference between the making of twine and cord and rope, as far as I know it's chiefly a matter of scale.
I would think that a spinning wheel could make light cord, which is basically a really sturdy yarn, which means that farmers could make their own twine and cord, at least in smallish quantities.

Commerical rope from colonial cordage factories, "rope walks," was made mostly from hemp. Farmers would make it out of whatever they had, mostly hemp or flax, but also wool or even hair, depending on its intended purpose.
For a laundry line or household use, probably hemp or flax.
There might have been rope from agave, jute, or abaca floating around from the ports, but if you're talking strictly local manufacture, I would go with hemp or flax.

Hemp rope/line was commercially available throughout the colonial Americas, jute was common in some ports and more developed areas. In areas without a commercial ropewalk (the colonial-era rope factory), flax was often used though it's inferior and not suitable for larger ropes or seafaring vessels. A simple cord might even be made out of cowhide.

Pretty much, hemp ruled the rope world for thousands of years, with flax being the poor-man's alternative.

Jeff