Q:

In the Christopher Nolan Batman Trilogy, how did they select the villains?

I was just curious if Christopher Nolan ever gave reasoning behind his choice of villains (Ra's, Joker, and Bane) to make appearances in the trilogy. I was thinking maybe the choices were more "iconic" to the series and would bring more butts to seats in the theatre, but it's hard to say Bane is more iconic than Mr. Freeze, the Riddler, etc. 
Disclaimer: I haven't read the comics or other canon, if it's something obvious like that was the order they appeared in the comic series. 

A:

Ra's al Ghul
It would seem that writer David Goyer had a great deal of influence over the selections of all the villains.

David S. Goyer, who co-wrote Batman Begins with director Christopher
  Nolan, says they wanted to broaden the parameters of the Batman
  mythology. "I think, of the Batman villains, Ra's Al Ghul is the most
  complex," he told Creative Screenwriting magazine. "We modelled him
  after Osama bin Laden. He's not crazy in the way that all the other
  Batman villains are. He's not bent on revenge; he's actually trying to
  heal the world. He's just doing it by very draconian means."

So Ra's al Ghul was chosen because he wasn't insane, but rather a Well-Intentioned Extremist. This was meant to strike a constrast with the conception of the insane Batman villain, exemplified by the Joker. 
The Joker
Wikipedia, referencing an unretrievable Empire article, suggests the Nolan wanted to reinterpret the Joker. I think this is probably the most obvious case, though. The Joker is the quintessential Batman villain. He is the logical choice for any series, and indeed was possibly going to appear in the third film before Heath Ledger's death. 
Bane
Bane was chosen because he was a brutal, physical villain, to contrast with the psychological games of the Joker:

With Bane, the physicality is the thing,” Nolan said. “With a good
  villain you need an archetype, you know, you need the extreme of some
  type of villainy. The Joker is obviously a particular archetype of
  diabolical, chaotic anarchy and has a devilish sense of humor. Bane,
  to me, is something we haven’t dealt with in the films. We wanted to
  do something very different in this film. He’s a primarily physical
  villain, he’s a classic movie monster in a way — but with a terrific
  brain. I think he’s a fascinating character.  I think people are going
  to get a kick out of what we’ve done with him.”

According to Wikipedia, referencing (again) a 2013 Empire article, the Riddler was initially considered for the role of antagonist, but rejected for the reasons previously mentioned, i.e. a lack of physicality and being too similar to the Joker.
Unsurprisingly, David Goyer was influential in choosing Bane. 
Looking at the process as a whole, it would seem that variety was the main goal.
Ra's al Ghul and Bane were both chosen because they broke the mold set by the Joker. 

