I found this article by Jonathan Turley interesting. He proposes setting the size of the U.S. Supreme Court at 19 members so that replacement of justices would be more regular (by making them more frequent), and justices would hold less individual power. It is well known that Associate Justice Anthony Kennedy is the one swing vote on the current court, because the other eight justices are polarized by ideology into two equal camps.
I like the idea of increasing the size of the court for the most part, although it feels weird that it should even have to have more than one person on it. What's right is right. It shouldn't be about a weighing of opinions, but a determination of fact. As a logical person, I would expect that to be possible for one well qualified person to do. But that is not how the world works, so we need to dilute individual power.
Now, rather than focus the process on (gradually) increasing the size of the court, which requires that we either change the law more than once or lay out a 10-to-20-year timeline, I would enact a system that deals more directly with evening the Presidents' influence on the court, and does so with a one-time change in the law regarding appointments to the court.
Let the President appoint one person to the Supreme Court at the beginning of his term. This allows the people's choice of leader to have immediate effect on the character of the court, and it evens out the relative influence of the Presidents. (Think of the fact that Jimmy Carter never appointed anyone to the Supreme Court and the five terms of Republicans that surrounded him in history appointed ten, more than replacing the entire court.)
Of course, if Presidents got one freebie and replaced everyone that left the court, it would eventually be too large. So some of the vacancies need to go unfilled and simply disappear. Justices have averaged about 18-year reigns, perfectly requiring our nine-member court to change one person about every two years--two replacements (on average) per Presidential term. If we wanted to keep things that way, the rules would be like this: The President gets his inaugural freebie, and because of that, doesn't get to replace the first vacancy that arises during the term. The third vacancy would also go unfilled. So if no vacancies arise, the court gets one member bigger, if one or two vacancies arise, it stays the same, and if three or more vacancies arise, it gets one member smaller. Since two vacancies is average, the nine-member court that we start with would generally drift back to that number.
If we want the court to grow in membership, we would just set the unfilled vacancy to a higher number, the fourth or fifth of the term. If the 18-year reign continues to hold true, setting it to the fourth vacancy could lead the court to hover around 13-1/2 members, and the fifth vacancy around 18. And the longer the justices sit, the more company they'll have. If it turns out that the court doesn't attain a nice size, we can adjust the vacancy threshold again. It's as simple as that.