Well, this post covers lots of ground - on which there's certainly room for various opinions.
Among maritime museum professionals and long-time sailing ship models the general concensus seems to be that the best rigging material is linen. Modelers have contended for decades that it's the most durable and authentic-looking material there is.
I personally have some doubts about that argument. I've seen linen on fairly old models (the Crabtree collection at the Mariners' Museum, for instance) that hadn't held up as well as expected. I've also seen other fibers that have lasted longer. In my opinion the care a model gets and the environment in which it resides are far more important to its survival than the material used for the rigging.
My personal favorite for a long time has been silk. Some people contend it doesn't last as long as linen, but I question that. I've got some spools of silk that I bought 25 years ago, and seem fine. Other spools that I bought at the same time are brittle; the thread snaps if you give it a good yank. I can only speculate that the difference has something to do with the dye that was applied to them (in some cases by me). Two of my models that I rigged with silk are now more than 20 years old, and the rigging literally looks as good as new. They've lived in plexiglas cases, out of direct sunlight.
In practical terms, unless you're willing to take some pains to order stuff over the web (and dye it and probably spin it up into the desired diameters yourself), this is largely moot, because both silk and linen are extremely hard to find. Bluejacket sells several diameters of linen (in white and black - none of them very fine), and I've found a couple of sites that sell silk (though I've never ordered from them). Most of the stuff sold by ship modeling suppliers nowadays is either cotton or synthetic.
Cotton is pretty flabby stuff, and seems to be hygroscopic (i.e., it shrinks and expands with changes in the humidity). Some veterans distrust synthetic thread on the grounds that it deteriorates in the presence of some atmospheric polutants. I honestly don't know about that - and synthetics haven't been around long enough to provide definitive evidence.
I really wish some ship modeler who has a day job in the field of chemistry or physics would undertake a genuine, scientific study of this subject. I suspect a lot of what's been written about it is hogwash.
Model Shipways (via <www.modelexpoonline.com>) sells some synthetic rigging line that I like; I used it on my last model. It has a nice, rope-like texture and comes in quite a few sizes.
There's also some doubt and controversy about color. Here's what I have to offer, based on the particular subjects I've read about.
Rigging line purchased by the British Navy in the eighteenth and early nineteenth centuries was hemp, and was supposed to be soaked in Stockholm tar as part of the manufacturing process. I've never seen Stockholm tar, but it's widely described as being of a rich, medium brown color. That's what I use for running rigging on models of that sort. Standing rigging was (I think) additionally treated with a surface coating made from tar and lampblack, which, if it wasn't pure black, must have been pretty close. Most serious ship modelers make their running rigging brown and their standing rigging either very dark brown or black.
For periods prior to the eighteenth century I don't know much about rigging color. I'm not sure when the practice of tarring the standing rigging came in. Most of the old "Admiralty" (more properly "Navy Board" models) from the seventeenth and early eighteenth centuries have rigging that's all the same color. If it's original it's pretty dark by now, due to age; it probably was originally the dull brown or beige color of undyed linen.
In the latter part of the nineteenth century, standing rigging began to be made of iron or steel wire, often (though by no means always) treated with a concoction with tar in it that would have made the color black (or nearly so).
An accurate model has several different diameters of rigging line. There are some good books on the subject that are helpful here; if you'll be more specific about the time period of your ship, maybe I can make some suggestions. The general (though far from universal) rule is: the higher up, the smaller the diameter.
If you do look up rigging sizes in a table, remember that rope is usually sized according to circumference - not diameter. The circumference (or so my high school geometry teacher told me) is equal to the diameter times pi (which equals 3.1416 - or 3, for practical purposes). So if you're reading rigging sizes from a table and your model is on the scale of 1/200, you need to divide the circumference of the full-size rope by pi and then by 200 to determine the diameter of the thread you want to use. To make life simple, divide the number in the table by 600 and you'll be close enough.
Two golden rules about rigging models: 1. If in doubt about color, err on the dark side. 2. If in doubt about diameter, err on the small side. Overly-fat, too-light-colored thread can wreck the scale appearance of a model faster than almost anything else.
You probably wish by now that you'd never asked these questions. As I said, this is a big, complex topic. Hope this has helped a little. Good luck.