The source of the issue is something called the Emoluments Clause, which is a clause in the Constitution that essentially states that, while a person is serving a term of office to which they were elected, he or she may not be appointed to any civil office which was newly created, or for which the compensation was increased in that same term. So because there was a pay raise given to the position of Secretary of State during Senator Clinton's current term, she is ineligible for appointment to this office.
The wording: "No Senator or Representative shall, during the Time for which he was elected, be appointed to any civil Office under the Authority of the United States which shall have been created, or the Emoluments whereof shall have been increased during such time; and no Person holding any Office under the United States, shall be a Member of either House during his Continuance in Office."
That's not all, either. This is not, apparently, the first time the issue has come up. Multiple examples are available at the sources above, but one in particular would be pertinent in the case of Hillary. When President Nixon wanted Sen. William Saxbe for his Attorney General, he had Congress reduce the wage, which had been increased and triggered the clause, in order to get around the restriction. The maneuver has been dubbed the "Saxbe fix".
The DailyKos article urges action soon, seeming to favor a Saxbe fix. The Volokh article features two legal experts who each arrive at a different conclusion, one favoring a Saxbe fix, one opposed. The grounds of their two positions are thick in the style of attorneys writing to attorneys, but worth reading anyway.
One thing does seem clear. It is, as John O'Connor put it, "beyond dispute that Senator Clinton is currently ineligible for appointment as secretary of State." After reading three articles on the subject, thick though they are with legalese, it's hard to argue with that assessment. Hillary Clinton is, under the emoluments clause, currently ineligible for the office of Secretary of State. What will be done about that status remains to be seen.
Update I: More from Volokh, an email from Constitutional Law expert Prof. Michael Stokes Paulsen is available here. Excerpt:
Thanks for alerting me to this fascinating (and fun) issue! I've played in this particular sandbox before [as to Lloyd Bentsen], and am amused to see it return in slightly different form.
So, "Is Hillary Clinton Unconstitutional?" In a word, Yes -- or, to be more precise, a Secretary of State Hillary Clinton would be unconstitutional.
The Emoluments Clause of Article I, section 6 provides "No Senator or Representative shall, during the Time for which he was elected, be appointed to any civil Office under the Authority of the United States, which shall have been created, or the Emoluments whereof shall have been encreased during such time." As I understand it, President Bush's executive order from earlier this year "encreased" the "Emoluments" (salary) of the office of Secretary of State. Last I checked, Hillary Clinton was an elected Senator from New York at the time. Were she to be appointed to the civil Office of Secretary of State, she would be being appointed to an office for which "the Emoluments whereof shall have been encreased" during the time for which she was elected to serve as Senator. The plain language of the Emoluments Clause would thus appear to bar her appointment ... if the Constitution is taken seriously (which it more than occasionally isn't on these matters, of course).
Update II: More details from the Washington Post