Read the book. The movie doesn't do justice to the real science behind it.
This. I liked the movie, but they had to simplify some of the more eye-opening statistical revelations because they wouldn't translate well to an actual narrative.
For one, I think the OP mischaracterizes Beane's and dePodesta's system by claiming that they "didn't go for scoring". In fact, what they found was that fielding ability is vastly overrated, because in baseball, defense depends so much on one player, the pitcher, such that everyone else's contributions pales in comparison. Every batter, on the other hand, is almost as important as any other, with only minor variations depending on placement within the rotation. They were able to replace a truly all-star talent on the outfield by picking up a couple of glorified designated hitters and sticking them in field positions where they'd do the least damage, including putting a former catcher who couldn't even throw the ball anymore and playing him on first base. (That part wasn't a Hollywood embellishment, that really happened.) They couldn't replace Damon's defense, but they could replace his Run Differential contribution over the course of a season by adding offense on the cheap, and in baseball offense is easier to add cheaply than defense. (For one thing, fielding ability deteriorates quicker than batting ability in over-the-hill players.)
In hockey, I think it's the reverse; defense is more cost-efficient than offense. So as much as many, including myself, may hate it, the "systems hockey" (i.e. boring defensive hockey) of Tippett and Hitchcock may be the closest equivalent. And the closest there's been to the A's ability to take bargain bin players and reaches at the draft and turn them into superstars is some teams' ability to make any goaltender look Vezina-worthy. We really haven't seen a team that's able to do that with skaters.
Some s.abermetric-style statistics I'd like to see used in hockey, even if I have no clue how they'd be reliably tabulated:
(1) the ability to make the opposing team take long shifts on defense. This is the equivalent of the pitches per at bat statistic that the A's valued, because it demonstrated the abilities to both control the strike zone and wear out pitchers.
(2) shots through in traffic
(3) rebounds/shot
(4) Corsi dropoff per ten seconds after thirty seconds (measures ability to take long shifts; good for evaluating potential penalty killers)
(5) clearing percentage (what percent of attempted clearings from the defensive zone actually result in clean, legal clearings)
(6) pass percentage in the neutral+offensive zones
(7) pass reception percentage in the neutral+offensive zones
(8) ability to either shoot off or establish control of bouncing, rolling, overly fast, or misaimed pucks
I don't know much about hockey superstats yet (other than QCo), some of these may already exist.