Reading through Genghis Khan and the Making of the Modern World and it says, "Mongol officials sent their Korean subjects a demand for one hundred thousand sheets of their famous paper." This is the first I've heard about Korean paper being special. What made it so famous?
Korean paper, at least the ones exported to China (there were at least 15 or 16 varieties at the time) was simply high-quality, and fairly popular in China. There are various Chinese records about Korean paper at the time, but it appears to have been popular because it was white, shiny, and durable, and one Chinese writer said that high-quality Korean paper was as good as the best Chinese paper. Emperor Zhangzong of Jin supposedly used Korean paper, and the cover of the original copy of the Yuan Shi was Korean paper.