What does “double-down” mean when used in a context other than Blackjack?
Solution 1:
A fuller quote from that article:
Bachmann, at least so far in the campaign, has shown a willingness to acknowledge mistakes she makes on the campaign trail. Palin is notorious for doubling-down on perceived missteps, typically blaming a biased media for not telling the whole story.
Since he contrasts "acknowledgement" with "doubling-down", it seems clear that Chris Cillizza (the political analyst here) is saying that Palin does not easily acknowledge mistakes. He unfortunately used a term which is rare enough that it is not in dictionaries.
In such cases, we go by context to understand the term. "Doubling-down on" means here that Palin wants to maintain face, and will sometimes not admit when she made a mistake.
Solution 2:
On another site I found someone quoting the OED on the phrase:
intr. Pontoon (Blackjack). to double down: to double the bet after one has seen the initial cards, with the requirement that one and only one additional card be drawn. Also in extended use: to engage in risky behaviour, esp. when one is already in a dangerous situation.
So in this case, Palin, already in a vulnerable position because of her "missteps," is taking a risk by blaming others.
Solution 3:
If we consider the context
Several readers point out another key difference between Bachmann and Palin — their approach to criticism. Bachmann, at least so far in the campaign, has shown a willingness to acknowledge mistakes she makes on the campaign trail. Palin is notorious for doubling-down on perceived missteps, typically blaming a biased media for not telling the whole story.
I think the meaning is clearer. Palin not only makes political mistakes, but then further compounds those blunders by blaming the media for misrepresenting her, instead of admitting that she said something nonsensical about Paul Revere.
It's a gamble to blame the media, not only because the media is the outlet for her message, but also because if people don't believe the media is to blame, then she looks even worse.
To answer your other question, I have occasionally heard double-down to refer to a situation where someone takes a bold risk when they are already gambling. For instance, I had not studied for an exam and I doubled-down by sleeping until fifteen minutes before class.