This article at the MIT Sloan School of Management summarizes a research paper by University of Chicago public policy professors Christopher R. Berry and Anthony Fowler entitled "How Much Do Coaches Matter?" Berry and Fowler found that in all sports, coaches "explain about 20 to 30 percent of the variation in a team’s success."
Specifically germane to the NFL, Berry and Fowler found that coaches "significantly affect" a team’s fumbles and penalties. Also, "the researchers found that New England Patriots head coach Bill Belichick was 18 percent more likely to win a game (based on the quality of his opponent and home field advantage) than an average coach. While that isn’t the highest percentage of his peers, it is 'extremely unusual' given his 17 seasons [as measured in the study] with the team." Since Belichick was hired as head coach in 2000, the Patriots have the fewest turnovers in the league, and they have committed the fifth-fewest penalties.