Matt Yglesias supports 'em. I think it's an interesting idea. Looking around, I see that there's a literature surrounding it, one that I haven't read.* The leading proposal is for a staggered single, eighteen-year terms, meaning that there would be a new nomination every other year.
As I've said, I tend to be very conservative about institutional design. I'm suspicious of Seligism -- Bud Selig, the current baseball commissioner, is constantly supporting changing long-standing design because some minor flaw turned up, without stopping to consider how various portions of the design are interrelated, or that minor flaws are inevitable regardless of design. I'm even more suspicious of those who turn frustrations with losing in a democracy into enthusiasm for changing the system. On the other hand, I'm not against all reform. Serious institutional breakdown, especially with a good case for inherent design flaws, should be met with reform -- the current most obvious case within American politics is California, with its impossible budget politics and destructive initiative process. Another reason to support reform is when the underlying reality that the rules are designed for changes, so that stable rules yield an unstable political system.
One can make a pretty good argument, I think, that lifetime appointments for Supreme Court Justices fit that last category. As Linda Greenhouse reported back in 2007, the actual length of terms for Justices has gone up dramatically since 1970 (from an average of about 15 years before then to an average of about 26 years since). Since life expectancy continues to grow, I expect that number to only continue to increase in the future. And the longer each appointment lasts, the more valuable it becomes, which pushes presidents to choose younger and younger nominees (although I think we're fairly close to the lower limit on that). Elena Kagan, should she be confirmed, may well be on the court for forty years. At some point, I think that's a very different system than that the Constitution envisioned.
As far as the advantages and disadvantages of the present system, Yglesias identifies two problems. The first one, which has to do with the process in which Justices are groomed and selected, doesn't really bother me much; as far as I can tell, the most recent selections under Bill Clinton, George W. Bush, and Barack Obama are all highly qualified and capable, so I'm not really worried about what they had to do to get there or about those who were blocked because the system works against them. It is true, however, that the high value of each pick creates incentives to selected a certain kind of nominee: young, ideologically reliable, and ambitious enough for the Court that he or she has known from an early age to avoid saying potentially controversial things.
The second problem is that the system produces a lot of random results. I do think is a serious flaw. There's no reason that the election of Jimmy Carter (who had no Court openings) should be worth less than George H.W. Bush. Nor is it reasonable that a president can be far more influential because his nominee lives another forty years, while that of another drops dead after a decade. Both of these always existed, but they matter far more when average tenure doubles.
The third problem is that it's not exactly clear why the results of elections decades ago should have so much influence in governing us today. Justice Stevens was nominated by President Ford (and for what it's worth, that's Unelected President Ford) and confirmed by an extremely liberal Senate. And then Americans elected Carter, and then Reagan, and then Bush, Clinton, Bush...and still, the dead hand of the 1972 presidential electorate and the 1970, 1972 and 1974 electorates and the Senators they chose controlled 1/9th of one of three coequal branches. Abortion is important, right? Roe vs. Wade was decided after Nixon was reelected, and after two-thirds of the Senators who confirmed Stevens were chosen. Of course, the nature of the Constitutional system, including not only lifetime judicial appointments but also long terms for the president and Senators, imply that the Framers intended at least some of our past decisions to govern the present and the future. But again, at some point a difference in degree becomes a difference in kind.
I think it's a good topic for debate...my guess would be that over time, the incentive for reform will get stronger as Justices serve for longer and longer terms. For better or worse, however, anything the requires a Constitutional amendment, as this surely would, is always going to have rough going, and that's far more true of those things that require a Constitutional amendment and would change the balance of power between the parties (in this case, on the Court). So while I think it's a promising idea, I think the odds of it being implemented are very low.
*Political science is funny that way. "American Politics" is my home subfield, and "institutions" within that, institutions are generally defined as the presidency and Congress for sure, parties-as-organizations probably, and perhaps the bureaucracy. Not, however, the courts. That's in a "Public Law" subfield. I try to keep up, but I follow far more of the popular discussion of courts-related stuff than I do the political science literatures.