Here are the relevant constitutional provisions, in relevant part:
- The Twelfth Amendment: “[N]o person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.”
- The Twenty-Second Amendment: “No person shall be elected to the office of the President more than twice ….”
- Article II, § 1, cl. 4: “No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained to the age of thirty five years, and been fourteen Years a resident within the United States.”
The question, then, is this: Does “constitutionally ineligible to the office of President” mean (A) “constitutionally barred from being elected to the office of President,” or (B) “constitutionally barred from serving in the office of President”?
If it means option A — if “eligible” is roughly synonymous, for elected offices, with “electable” — then Bill Clinton would be ineligible to the office of president because of the 22nd Amendment, and thus ineligible to the office of vice president because of the 12th Amendment. On the other hand, if “eligible” means simply “constitutionally barred from serving,” then the 22nd Amendment doesn’t speak to whether Bill Clinton is eligible for the office of president, since it only says that he may not be elected to that office. And because there’s nothing in the constitution that makes Clinton ineligible for the presidency, the 12th Amendment doesn’t make him ineligible for the vice presidency.
My tentative answer is that “eligible” roughly means “elected.” I realize that this is far from perfect evidence — it’s 40 years later than the usage — but the earliest law dictionary that I could find that contained the term, Bouvier’s (1843), defines “eligibility” as “capacity to be elected.” (I take it that, by extension, for appointed offices it would mean “capacity to be appointed.”) If that’s how the term was understood in 1804, then Clinton would not be eligible to the office of president, and thus under the 12th Amendment not eligible to the office of vice president.
Some mid- to late-1800s cases also define eligible as referring to “capacity of holding, as well as capacity of being elected to an office” (see Carson v. McPhetridge, 15 Ind. 331 (1860)); but that’s in the context of saying that someone who isn’t eligible to an office isn’t capable either of holding the office or being elected to it. I’ve seen no evidence that, contrary to the Bouvier’s definition, a person would have been seen in the early 1800s as being “eligible” to an office when he was legally barred from being elected or appointed to it, and the only question related to whether he could automatically assume it under some succession statute.
On the other hand, Bruce G. Peabody & Scott E. Gant, The Twice and Future President, 83 Minn. L. Rev. 565 (1999), argues the contrary, though I find myself tentatively unpersuaded by the article’s position:
First, it is by no means clear that the term “eligibility” as used in the Twelfth Amendment refers to or incorporates a person’s reeligibility under the Twenty-Second Amendment. At the time the Twelfth Amendment was written there was, of course, no Twenty-Second Amendment; therefore, the Twelfth Amendment could not have originally meant to preclude someone from being Vice President who had been elected President twice. Rather, the Twelfth Amendment’s reference to “eligibility” likely pointed only to the “eligibility” provision of Article II, Section 1, clause 4 ….Second, even if the 12th Amendment’s eligibility provision is to be read in light of the proscriptions of the 22nd Amendment, it could be read as affecting only persons who would become president. If this understanding is correct, the 12th Amendment’s provision that “[n]o person constitutionally ineligible to the office of President shall be eligible to that of Vice President of the United States” has no effect on individuals who might simply act as president. In other words, a vice president “constitutionally ineligible to the office of President” might occupy the vice presidency and eventually act as president, while being ineligible to assume that office by becoming president through succession….Third, and most importantly, even under the most expansive reading of what constitutional “eligibility” might include[,] … we do not believe [for reasons elaborated elsewhere in the article -EV] an already twice-elected President is “constitutionally ineligible to the office of President.” … Even if one leaves aside [scenarios involving succession from the vice presidency to the president], there are other non-electoral means of reassuming office available to a twice-elected president [– a person’s acting as president under succession statutes triggered by the unavailability of either the vice president or president, or becoming president if chosen by the House of Representatives when no candidate gets a majority of the electoral votes]. Thus, if the meaning of “eligibility” under the 12th Amendment was transformed with the adoption of the 22nd Amendment, the 22nd Amendment still does not render twice-elected presidents “constitutionally ineligible to the office of President,” and it therefore cannot be said that the 12th Amendment prohibits a twice-elected president from serving as vice president.
Of course, this is unlikely to get tested — generally speaking, a candidate wouldn’t want to select a running mate when there’s serious doubt about the running mate’s eligibility, and when there are many other good alternatives as to whom there’s no doubt.
UPDATE: From Akhil Reed Amar’s “America’s Constitution: A Biography” (2005), chapter 12:
Yet some ambiguities have remained to the present day. The Two Term Amendment’s opening words barred a person from being elected president more than twice. But what about a two-term incumbent who thereafter sought and won the vice presidency and then moved back into the Oval Office as a result of the new president’s death, disability, removal, or resignation? Would it matter if the resignation had been prearranged? What if our two-termer had subsequently become vice president not by being elected to that office but by appointment thereto under the later Presidential Succession Amendment? Even if all such vice-president-turned-president gambits were deemed to violate the Constitution’s letter and spirit, should the same broad reading apply if a two-termer later became, say, secretary of state and was then called upon to act as president in a double-vacancy scenario, pursuant to a valid congressional succession statute?
 Some scholars and pundits — most notably, Bruce Peabody and Scott Gant — have contended that such gambits remain constitutionally permissible. But this seems doubtful in light of the letter and spirit of the Twelfth Amendment: “no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President.” Peabody and Gant effectively treat the Twelfth’s command as beside the point. On their view, although a two-term incumbent cannot be again elected to the presidency, he is nonetheless eligible and therefore also eligible to be vice president. Bruce G. Peabody and Scott E. Gant, “The Twice and Future President: Constitutional Interstices and the Twenty-second Amendment,” Minnesota LR 83 (1999): 565, 619-20.But this reading slights the facts that the words “eligible” and “electable” spring from the same Latin root, and that standard dictionaries have long included “electable” as one of the standard definitions of “eligible.” See, e.g., OED entry on “eligible.” Given these facts, it would seem that a two-term incumbent is “ineligible” to the presidency (within the meaning of the Twelfth Amendment) precisely because he is made unelectable to that office (by the Twenty-second) — and is thus barred (by the Twelfth) from being elected to the vice presidency in the first place.The Twenty-second’s retroactivity rules would seem to confirm this reading of the Twelfth. These rules applied not only to a past incumbent who might in the future be “elected” president but also to one who might thenceforth “act as President” — paradigmatically, by being elected vice president and then moving back into the Oval Office via death, disability, or resignation. Arguably, this scenario might differ from the case of an elected vice president because no offending presidential or vice presidential “election” — as distinct from a mere “nominat[ion]” and “confirmation” under the Twenty-fifth Amendment — has occurred…. In this scenario, the eligibility rule of the Twelfth Amendment … simply does not apply. On the other hand, the current succession law, 3 U.S.C. 19(e), excludes from the line of succession any officer who is not “eligible to the office of President under the Constitution” and thus would seem to raise similar issues.