Well, first, why don't we look at what the Constitution says?
The judicial Power of the United States, shall be vested in one supreme Court, and in such inferior Courts as the Congress may from time to time ordain and establish. The Judges, both of the supreme and inferior Courts, shall hold their Offices during good Behaviour, and shall, at stated Times, receive for their Services, a Compensation, which shall not be diminished during their Continuance in Office.
The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution, the Laws of the United States, and Treaties made, or which shall be made, under their Authority;—to all Cases affecting Ambassadors, other public Ministers and Consuls;—to all Cases of admiralty and maritime Jurisdiction;—to Controversies to which the United States shall be a Party;— to Controversies between two or more States;--between a State and Citizens of another State;--between Citizens of different States;--between Citizens of the same State claiming Lands under Grants of different States, and between a State, or the Citizens thereof, and foreign States, Citizens or Subjects.
U.S. Const., Art. III.
So. . . life tenure is baked into Article III, which means one idea I have heard, limiting the terms of the justices, would require a Constitutional amendment. I think that's a non-starter. There have been hundreds of proposed amendments, of which a total of seventeen have made it through (the first ten were basically in the Constitution from the git-go).
A far more intriguing idea for "de-politicizing" the Court would be to expand it to fifteen seats, with ten seats filled by the President with the advice and consent of the Senate, and five seats to be filled either by judges on rotation from the Federal appeals courts (again, requires a Constitutional amendment), or by unanimous consent of the ten traditionally-appointed justices. This MAY be possible without an amendment.
Many people don't know this, but beyond the provisions of Section 1 of Article Three, everything about the Court, including the number of justices, is set by the Judiciary Act of 1790, as amended.
Personally, I think the easiest fix is to go back to the requirement of a 60-vote supermajority in the Senate to confirm. That seems to have worked pretty well in getting relatively politically moderate justices on the bench.