If they want to bring in Art Briles, it would only fuel the perception that Snyder got involved again and that RGIII became a favorite and now runs this team. I would go so far as to say even if Briles is by far and away the best guy for the job on merit, that they do not hire him.
The Redskins may also want a coach who can make personnel decisions. That hasn't worked out either. Steve Spurrier had autonomy in picking many of his personnel, that didn't work. Gibbs had autonomy, and what we got was a top heavy roster with no depth. Shanahan had it, and I think he got overwhelmed. Allen is a cap expert, not a talent evaluator. He needs to bring in that talent evaluator to be the GM, and then from there we find a head coach.
I would look for a young, promising coordinator. Someone who is a fresh, innovative mind, who can connect with RGIII and repair the relationship between player/coach. I would suggest that our primary targets be both Greg Roman and Darrell Bevell, of the 49ers and Seahawks respectively. Both are running offenses with similar quarterbacks right now, both have coached against RGIII and have a background knowledge and would be a fresh face for the franchise.
Honestly, that is what is most important, the fresh face. It equals a new start, and ever since Snyder took over we've brought in name coaches who have already had success elsewhere and they are unable to replicate it. For as much as I'd like to see Jon Gruden come in, it's the same story all over again.