How much brown sugar should I substitute for white sugar?
Dupe, but a good question. See Satanicpuppies answer here: Brown sugar instead of white sugar

I know you asked re: cranberry sauce specifically, but this is a great article regarding the differences in sugars, generically, which might help in the future:

"Sugar can be a single molecule made up of carbon, hydrogen, and oxygen—like glucose and fructose. Other sugars, like sucrose, or white table sugar, are made up of multiple molecules (in this case, one glucose and one fructose), tied together with chemical bonds. Now, when sucrose is heated with an acid, it breaks back down into the two smaller sugars, glucose and fructose, resulting in something we call invert sugar. This small chemical change makes a big difference: While sucrose is hygroscopic, invert sugar is even more water-loving.

And this can make a tremendous difference when, say, you want to bake a chewy, rather than crunchy, cookie."

http://www.americastestkitchenfeed.com/cooking-science/2012/10/we-prove-it-sugar-changes-texture-and-sweetness/