I would limit SCOTUS judges to a maximum of 20 years on the federal bench, or when they reach the age of 75, whichever comes first. Of course, doing so would require a constitutional amendment. (and if we're amending the constitution, we might as well go all the way and change it so voters can kick out federal judges)
One trend in American politics that makes little sense to me is that most SCOTUS judges are choosen from the ranks of existing federal appellate court justices. This includes conservative justices like Alito and Scalia. If this were Europe, I suppose that would make sense, but the way we choose Congressmen and Presidents tend to be the opposite -- more often that not we fill the office with someone from OUTSIDE Washington that has held office at the state level. Why not do the same with SCOTUS judges? People that have been in the beltway too long shouldn't get to stay there longer in a more powerful position.
If I was President, I'd come up with a short list by having my advisers vet existing state judges and giving me a list of the top 10 most reliably conservative justices on state supreme courts (Judge Roy Moore of the Alabama Supreme Court, etc.). Then I'd research them further and the interview the prospective choices until I narrowed it down to one.