"Voici un stylo pour que vous puissiez écrire votre nom."
Translation:Here is a pen so that you can write your name.
No, I agree that they are both superfluous in English. Both variations should be accepted as should "Here is a pen so that you can write your name". I may translate que and include "that" but I would never translate écrire to "write down" unless it was a sentence referring to dictation.