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.
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 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.