I've noticed Bill Cowher and Marty Schottenheimer being heavily mentioned both in the comments here and in discussions across the internet as desired replacements for Shanahan. They apparently bring experience to the table that is necessary to win in the NFL.
Schottenheimer's experience didn't help him accomplish anything of note in San Diego other than underachievement. Bill Cowher doesn't seem particularly interested in coaching in the NFL at the moment, why drag him back?
All that hiring a coach like Cowher or Schottenheimer would do is create the same kinds of problems that kept Shanahan around far too long, an unwillingness to question their decisions, a lack of accountability, and a burgeoning cult of personality.
Atlanta, Baltimore, and Miami brought in "inexperienced" head coaches to coach relatively unseasoned teams and all three had remarkable success this year.
Experience is overrated and shouldn't be anywhere near the top of desirable qualities in a new head coach.