Which sugar for brining?
For a brine, no it doesn't really matter, except that granulated sugar will dissolve more nicely than raw sugar.  The sugar is for flavor more than anything else, though it does help the brine some. It also promotes browning of the meat.  If you change one kind of sugar for another, I'd use a 1:1 swap by weight rather than volume so you don't have to worry about crystal sizes.

Though the sugar does help the brining action some, the salt is the important component.

From Cook's Illustrated:


  How does brining work? Brining promotes a change in the structure of
  the proteins in the muscle. The salt causes protein strands to become
  denatured, or unwound. This is the same process that occurs when
  proteins are exposed to heat, acid, or alcohol. When protein strands
  unwind, they get tangled up with one another, forming a matrix that
  traps water. Salt is commonly used to give processed meats a better
  texture. For example, hot dogs made without salt would be limp.
  
  In most cases, we add sugar to the brine. Sugar has little if any
  effect on the texture of the meat, but it does add flavor and promotes
  better browning of the skin.