I put this one on the other discussion, too (easy score)
From what we know about David in his early (and later) years, he was also an excellent kinnor (usually translated "harp") player. On
http://cdbaby.com/cd/mlevy ("King David's lyre, echoes from ancient Israel"), I found this:
In Biblical times, the Kinnor was usually made of cypress wood, or in very precious instruments, of sandalwood (I Kings 10: 12; described as "almug"). According to the ancient writings of the Jewish historian Josephus Flavius, who actually witnessed the Kinnor being played by the Levites in the Temple of Jerusalem, the Kinnor had ten strings, made of sheep gut.
Sheep gut strings fit well into the image of the young shepherd who uses materials from his (father's) livestock for his pastime - playing music and slinging. So, for his sling, wool would be a good candidate (I'm not a specialist, but I think that sheep's sinews are too short to be used as sling cords). I suppose either a woven pouch or, less probable, one out of goatskin. Sheepskin to me seems too soft and too thin to be effectively used for a sling pouch.