Which nuns get new names? I know that some Catholic nuns chose new names, like Mother Teresa (previously Agnes Gonxha Bojaxhiu), and others didn't, like Sister Helen Prejean.

What determines whether a woman changes her name on taking her vows? Is it up to the individual, or do different orders have their own particular rules or customs?
Name changes in the Bible indicate a significant change in the life of the person. For example,

Abram became Abraham. 


  "No longer shall your name be Abram, but your name shall be Abraham;
  for I have made you the father of a multitude of nations." (Gen 17:5)


Jacob became Israel. 


  Then he said, "Your name shall no more be called Jacob, but Israel,
  for you have striven with God and with men, and have prevailed." (Gen
  32:28)


Simon became Cephas, which means rock. 


  "So you are Simon the son of John? You shall be called Cephas" (which
  means Peter). (John 1:42)


Having said that, it is really up to the individual to take a new name. Usually, they like to take the name of a saint whom they are inspired by.