I realize it is impossible to say for sure what kind of cloth it was, and of course, we cannot be certain that this Biblical scene happened the way the author describes, but the cloth was likely a common fabric from the period. Cloth is an interesting component of material culture and has been used (in the Americas, the area I study) to reveal production, consumption, trade networks, and elements of home and domestic life. Those are actually what I am curious about. What kind of cloth was common in this part of the Roman Empire during the early first century? Where did it come from? How might it have been made or acquired by a poor family of common people? Who might have made it and how? How might it have ended up with that family? The very next verse in the story talks about the shepherds tending their flocks (of sheep? Goats?), perhaps they had a hand in it? Or maybe it was a fibrous plant of some kind used to make the cloth in that region at the time?
Cotton, linen,and wool would all have been common. (Jewish law forbids mixing wool with cotton or linen.) People may have woven their own on hanging looms, as was common in Europe from the neolithic until the early modern period. They may also have purchased it or received it as a gift. Obviously they had clothing - cloth was very common and accessible.