A recent poll showed that more Americans could name the seven dwarfs than the nine Supreme Court justices. (Well, duh, it's harder to remember nine things than seven.) However, the validity of the poll results may have been compromised by the number of respondents who referred to Justice Scalia as "Grumpy."
Sure, we live in a pop-culture world where writing a killer dissent in Citizens United isn't going to make you as popular as Britney Spears—but writing that killer dissent will make you more popular than Britney with certain people. To be more specific, smart people who follow current events and care about our society and its foundations.
There is certainly some degree of fame associated with a position on the Supreme Court, and the justices are public figures to a large extent. They are followed by the press, and all of their foibles and eccentricities are well documented. Although the fame involved with a Supreme Court position is very different than pop-culture fame, this position isn’t for those who cherish privacy or worry about the skeletons in their closets.