Professor Matters for Independent Study?
In some ways everything matters, but probably not as much as you think. Probably not enough to worry much about it. A lot of things will help you, but working hard and gaining skills is pretty high up the list. 

I'd suggest that rather than trying to game the system, find courses (including independent studies) that will make you well educated and well rounded. There are fine teachers without a huge reputation. There are terrible teachers with a great reputation. There are all of the other combinations as well. 

Study something meaningful with someone who can help you learn it well. Work hard at it. The rest will follow.