Embarrassed to say I have only seen the Canyon once. Three and a half months before I moved, I decided I'd better go... since who knew if I'd ever get back. We went over President's Day weekend. It was the dead of winter, but absolutely beautiful, a true winter wonderland. Unlike here, AZ closes roads to drivers when it snows and the roads are in rural areas. We barely made it out of the park and back to Flag before the road into the park was closed. It was only the second time my kids had seen snow. We have a priceless video of my older son slipping on ice and making a quick recovery, on the way to the overlook. We had to explain that you have to watch the ground and be careful when there is ice. He just had no conception of this, as a desert rat.
I'd bring the same clothes you'd bring for x country skiing or hiking in any winter climate. Just make sure you have removable top layers.
Several of the teachers I worked with used to hike the canyon every spring break. I always was asked to come, as they knew I was a fitness instructor, worked out. But, back then, any outdoor endeavor scared the crap out of me. I seriously thought they would have to call that rescue helicopter for me. I *should* have just trained and done it, because now, I would love to.