Manny Ramirez is a clear upgrade for the White Sox, worth an extra win over Mark Kotsay (the current, impotent DH option for Chicago) over the remainder of the season, and a much better option for leveraged at-bats late in a close game.
There's some chance that Chicago will get more out of him than the Dodgers did because he'll never have to put on a glove -- which is a good thing, because at this point in his career he's among the worst defensive outfielders in baseball -- and it's possible that the move to a good home run park will produce better-than-expected results. It doesn't make the White Sox, four behind Minnesota in the loss column, sudden favorites to win the AL Central, but the move makes them more competitive against a slightly superior Twins team.
Although there will probably be a lot of hand-wringing over Ramirez's makeup and behavior, the more legitimate concern for me is whether his stat line this year is a function of playing in the weaker league and facing a pretty harmless basket of pitchers. Despite playing in the NL West, Manny didn't face Mat Latos or Tim Lincecum, had just one PA against Matt Cain, and just three against Ubaldo Jimenez, while he's done a lot of his damage this year against pitchers with fringe-average fastballs. His bat speed has slowed since his return from suspension in 2009, and I can't look at his strong .311/.405/.510 (AVG/OBP/SLG) line as evidence that it's back because he's done so much damage against the Rodrigo Lopezes of the world.
The main beneficiary in L.A. should be Trayvon Robinson, a slow-developing toolsy outfielder who broke out in the Cal League last year and actually raised his walk rate with a promotion to AA this year. The switch-hitting Robinson has above-average speed and quick hands with good hip rotation through his swing, although he tends to get out on his front foot a little early, cutting off some of that power. He played center all year in Jacksonville and would be plus in either corner outfield position. Whether he's ready to hit major league pitching right now is an open question, but the continually improving approach and good bat speed are both positive indicators, and it would be great for the Dodgers to give him some real playing time in September to see if he could break camp with the team next April.
As for the money saved, the Dodgers' alternative was supposedly third baseman Jon Gilmore, a fringe prospect at best, and I can understand their decision to take the money. But wouldn't it have been nice to give half that $4 million to Logan White and Tim Hallgren two weeks ago so they could have signed their sixth-round pick, LSU-bound right-hander Kevin Gausman?