It seems eligibility is in the news today. Via Instapundit, I read two articles, one at DailyKos and one at The Volokh conspiracy, regarding Hillary Clinton's eligibility to serve as Secretary of State for the Obama administration. My reading of the story online indicates the DailyKos blog came first.
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."