He replaces John Anderson, who didn't return after two years with the Coyotes.
The 51-year-old Brown spent the past three seasons with Vancouver, working on a staff that helped lead the Canucks to the Presidents' Trophy in consecutive seasons and a trip to the 2011 Stanley Cup finals.
Brown also has served as an assistant coach for Anaheim, Columbus and Chicago in an NHL career that dates to the 1996-97 season. He also was the head coach for four seasons with Adirondack, the Detroit Red Wings' AHL affiliate.
I got a chance to play a thousand games and that's pretty special to me. To get a thousand [points], it's a great accomplishment. I'm not going to hide my feelings, I'm proud of that. To do it on a win, do it on a goal, I think it makes it special.
— New York Rangers forward Martin St. Louis after scoring his 1000th NHL point on Friday