What did Brown do?; the Weight of and Opinion.
Cass Sunstein reflects on various responses to Brown v. Board of Education in the most recent New Yorker.

Not everyone thinks that it has aged well. Many progressives now argue that its importance has been greatly overstated-that social forces and political pressures, far more than federal judges, were responsible for the demise of segregation. Certainly, Brown has disappointed those who hoped that it would give black Americans equal educational opportunities. Some scholars on the left even question whether Brown was rightly decided. The experience of the past half century suggests that the Court cannot produce social reform on its own, and that judges are unlikely to challenge an established social consensus. But experience has also underlined Brown's enduring importance.

So what did Brown do? And more generally, what do the Court's decisions do?
Quickly, we can note the small, actual effect of Brown- and Sunstein gets at this in the article. Six years after Brown, none of the 1.4 million black children in the Deep South (Alabama, Georgia, Louisiana, Mississippi, and South Carolina) attended racially mixed schools. In the Brown II decision (a year after the first), the Court demanded that segregated schools move in "all deliberate speed" to integrate...a motion soon revealed as pedestrian rather than sports car.
In lieu of direct causation, Sunstein notes (citing some recent work) the catalyst theory:

But Klarman doesn't claim that Brown was irrelevant to the desegregation struggle. In his view, the decision catalyzed the passage of civil-rights legislation by, in effect, heightening the contradictions: inspiring Southern blacks to challenge segregation-and Southern whites to defend it-more aggressively than they otherwise would have. Before Brown, he shows, Southern politics was dominated by moderate Democrats, who generally downplayed racial conflicts. The Brown ruling radicalized Southern politics practically overnight, and in a way that has had lasting consequences for American politics.
...
Because "the post-Brown racial fanaticism of southern politics produced a situation that was ripe for violence," he writes, Northerners soon found themselves outraged by televised scenes of police brutality against peaceful black demonstrators. The civil-rights legislation of the sixties, including the very laws that led to the enforcement of Brown, arose from a sort of backlash to the backlash. Given these complicated causal chains, how important to our civil-rights history, in the end, was Chief Justice Vinson's fatal heart attack? Not very, in Klarman's accounting: "Deep background forces"-notably, the experience of the Second World War and the encounter with Nazi racial ideology-"ensured that the United States would experience a racial reform movement regardless of what the Supreme Court did or did not do."

All this brings me to a favorite topic: what power do (and more importantly, should) the courts have to shape social norms? The last lines quoted above suggest Brown was only so many words with little to no causal relation to civil rights. This might be so- as the empirical evidence of its effect can illustrate.

Then again, perhaps Brown is among the most important elements to the painfully slow civil rights struggle that persists today. And we have empirical evidence too: Brown is one of the most popularly regarded cases. People evoke it all the time. It doesn't matter whether they evoke the correct legal positions, nor whether they duly cite Brown II and the non-response Brown accomplished.

(I tend to think Brown did shape some legal aspects to civil rights. The equality principle argued over in the case had been in existence for close to a hundred years- but still needed, as it does today, some work. The Court wrote that separate "educational facilities are inherently unequal;" thus violating the equal-protection clause of the Fourteenth Amendment. This is significant beyond the feel-good equality principle. Plessy v. Ferguson (overturned by Brown) also demanded equality. It just allowed that equality to reside in separate quarters. Brown made the sociological comment that this doesn't work (although that position is still argued by, amongst others, historically black colleges.))

Back to theory, and away from Brown, some questions:
1) Can the Court shape social norms?
While majoriy decisions are not going to change minds on big social questions (homophobes aren't healed of that problem via Kennedy's Lawrence v. Texas decision), I think the decisions play a rather large role in shaping national norms.

2) Should the Court shape social norms?

yes