This article seems pretty interesting: Ruth Colker, Justice Sandra Day O'Connor's Friends, forthcoming in the Ohio State Law Journal. Here's a summary:

In an upcoming law journal article, Ohio State University Law Professor Ruth Colker examines how amicus briefs might influence judicial behavior.

Colker analyzed the 106 constitutional law cases heard by the Court during the tenure of Justice Sandra Day O'Connor (1981 to 2006) in which amicus briefs were filed. Colker focused on O'Connor's citation of amicus briefs because her recent departure from the Court offered a complete set of opinions spanning three decades and because her role as a swing voter makes it likely that amicus authors often had her in mind, "Hence an examination of her opinions in cases in which amicus curiae filed briefs can lend insight into whether amicus briefs have much influence on the Court's decision."

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Colker found that amicus briefs were most likely to be cited by O'Connor if they offered specialized facts in support of the majority's legal theory. In addition, Colker found that O'Connor was more likely to cite briefs filed by the Solicitor General, prestigious associations and the states. And she never cited briefs from organizations associated with overtly political causes -- on the left and the right -- such as the American Civil Liberties Union.

Interestingly, O'Connor was more likely to cite amicus briefs when she disagreed with their legal position than when she agreed with their position.

"If O'Connor is reflective of the Court, in general, then authors of amicus briefs, who wish to be cited by the Court, might want to focus more of their attention on factual development. They also might want to affiliate themselves with a neutral-sounding professional organization."

Of course, citation is different from influence (as the article is quick to note). Some briefs are influential but go uncited, and others are cited but have no influence.