This is going to be a disappointing answer.
They're made from some fabric unique to Lórien; from Fellowship (emphasis mine):
The Elves next unwrapped and gave to each of the Company the clothes they had brought. For each they had provided a hood and cloak, made according to his size, of the light but warm silken stuff that the Galadhrim wove.
[...]
They are fair garments, and the web is good, for it was made in this land.
Fellowship of the Ring Book II Chapter 8: "Farewell to Lórien"
And that's the most complete description given by Tolkien, in any of his writings.
As I see it, there are effectively two possibilities:
The cloaks are woven from some typical Middle-earth material, but woven with such Elvish skill that it becomes unusually light and warm1. In which case, the most plausible fabric is a wool or silk, both of which are known to the rest of Middle-earth
The material is unique to Lothlórien. In this case, it could be anything from cotton (which doesn't seem to exist as a textile in the rest of Middle-earth) to some mythical substance that doesn't exist in our world
Either possibility is as likely as the other, really.
An interesting thing to note2 are the similarities between the description of the cloaks, from above, and the Elven-rope given to Sam:
Three small grey boats had been made ready for the travellers, and in these the Elves stowed their goods. And they added also coils of rope, three to each boat. Slender they looked, but strong, silken to the touch, grey of hue like the elven-cloaks.
Fellowship of the Ring Book II Chapter 8: "Farewell to Lórien"
While there's no positive indication that the ropes and cloaks are made from the same fabric, it seems exceedingly plausible. One of the Galadhrim later says that the rope is made from something called "hithlain." Unfortunately, we also don't know what hithlain is made from; the name maybe translates to "holy mist"3, which isn't very helpful from a materials design perspective.
In the movies
Although not strictly relevant, I found this interesting4 and I wanted to share.
The cloaks used in the films were made from a special grey wool, apparently unique to a New Zealand company called Stansborough Fibres:
Their flock of blue grey sheep, now have wool that is unusually silky, soft and lustrous. This is the result of many years of selective breeding and hard work, producing a 'Stansborough Grey' flock of over twelve hundred animals, the only ones of their kind in the world.
Most accounts I can find, such as this one from a Lord of the Rings costume resource site, claim that the 'Stansborough Grey' fabric matches Tolkien's description rather well:
The beautiful weave, soft silky texture, natural hues and its ability to warm, despite its lightweight feel, exactly match the description given by Tolkien.
1 Note that I'm considering the properties of the textile independently of the magical effects of the cloaks; that is definitely a function of the weaver
2 Nod to maguirenumber6 for reminding me to add this in
3 from The Silmarillion appendix, hith means "mist". I'm not sure about the "lain", but it's possible that it related to "ain", which means "holy" (as in "Ainur", for example).
4 Keep in mind that I am a bit weird. Your interest may vary