I was always under the impression that the Chief Justice of the Supreme Court was a largely titular office - he/she wasn’t really supposed to have any more or less influence on decisions, opinions, etc. Why, then, don’t the Supremems pick the Chief Justice from their own ranks instead of having it exist as a separate position to which one may be appointed by the President?
The Chief Justice has a little more power than the others in that he controls certain administrative functions like who writes the majority opinion and he can extend an attorney’s time. The office is one picked by the President because it is one of the checks and balances designed in our constitution to keep each branch from having too much power.
Well, for one thing, he’s the Chief Justice of the United States, not just of the Supreme Court. He is the head of the entire Judicial branch of the U.S. government. I suppose that in order to maintain the balance of power between the branches, it was decided to have the choice of Chief Justice left up to the other two branches. When he votes with the majority, he assigns the writing of the majority opinion. CJ Burger was infamous for voting against his conscience if he was sure to lose anyway, just so he could pen the majority opinion himself.
The CJ has little actual power over the Judicial branch as a whole, but he has certain administrative duties. I think he can reassign circuit judges to other circuits, and I know he presides over the Judicial Conference of the United States, which meets once a year and concerns itself with workings of the system. As we have seen, he also presides over the Senate trial of an impeached President.
Rehnquist is a bit of an unusual case. Most Chief Justices have not been chosen from the ranks of the sitting Supreme Court justices. This is because picking a Chief Justice from the ranks requires two confirmations by the Senate. If when Burger retired, Reagan had appointed Scalia to the CJ post, it would have required only one Senate confirmation. But on the other hand, there is no guarantee Scalia would have been approved by the Senate as the CJ.
I got a few details wrong about the Judicial Conference. From a speech by CJ Rehnquist:
As far as reassigning circuit court judges, I’m not sure I got that right. It could be that he can reassign the circuit assignments of the Supreme Court Justices. Or something like that. It’s been a while since I read this.
To answer the OP:
The Constitution is silent upon the composition of the Supreme Court. Article III, Sec. 1 states only that there shall be at a minimum a Supreme Court (Congress is empowered to create such inferior courts as it deems needed). It also sets forth that federal judges get to hold their offices ‘during good Behaviour’ (they get to be judges for life unless impeached). Article II, Section 2 gives the President, with the Advice and by the Consent of the Senate, the power to appoint the Supreme Court’s judges.
Congress immediately established the composition of the Court by legislation. Note that the Constitution is silent as to who gets to set the composition of the Supreme Court. This isn’t an academic issue; if the Congress can set the number of justices, then this power can be used to apply pressure on the suppoedly seperate branch of the Judiciary. Indeed, on at least three occaisions, Congress has attempted to influence the politics of the Court by changing the number of justices, most recently during the 1930’s (the infamous ‘court-packing’ scheme). My recollection was that it had five justices at first, with Justice John Jay the Chief Justice.
Which brings us to the OP’s question. In setting up the composition of the Court, the legislation specified that there would be a Chief Justice and x number of associate justices. Currently, 28 U.S.C. §1 reads:
So, in answer to the question, there is a Chief Justice who is chosen by the President because the Congress set the court up that way, and that’s how it has been done ever since. 
I don’t think we’ve hit on it yet. The Constitution does not speak to the administration or offices of the Court. Article III, Section 1 merely says that there shall be a Supreme Court.
Thanks, Counsel. You posted while I was typing my follow-up.
I’m familiar with FDR’s court packing scheme (“the nine old men” as he liked to call them), I just wasn’t sure if the composition of the Court was legislatively mandated. Does the USC get so specific as to regulate how the CJ is chosen? The section cited merely says there shall be a CJ and AJs, and does not address the crux of my question - If the CJ is really primus enter partes, what’s the rationale for not having the AJs pick the CJ from among their own ranks? I can see it now, “Best two out of three falls for the robe with the gold stripes and the gavel.”