Why the surname 유 (Yoo) is translated as 刘 instead of 余 or 于？
I don't think it is translated as much as it is that every Korean family actually has their own Hanja.

According to Wikipedia's List of South Korean surnames by prevalence


  유


can be one of four Chinese characters:


  柳, 劉, 兪, 庾


Most Koreans have Hanja names &amp; they would certainly know which their surname is.



If you check out Wikipedia's Appendix:Korean surnames you can see that


  여


is the Southern Korean equivalent of


  余


while


  우


equates to 


  于, 禹