1) Declare the debt ceiling unconstitutional and keep on borrowing. Bruce Bartlett, Bill Clnton, and I support this one, based on Section 4 of the 14th Amendment. If it held, as it would be challenged in the courts, it would effectivly abolish this uniquely idiotic device. OTOH, aside from serious people like Laurence Tribe who say the debt ceiling is constitutional, Obama would certainly face impeachment by the House, if not removal by the Senate, and the financial markets might demand higher interest rates on US securities due to the uncertain legal foundation of any new borrowings. While his press secretary has supposedly ruled this out, Obama himself has never specifcally commented on this issue, indeed, has refused to do so. Non-trivial possibility he might follow this one, if he has the chutzpah.
2) The full haircut. Under the constitution the president (and the treasury secretary acting on his behalf) does not have the right to decide to pay some bills and not others (although this has been done in the past during hilariously labeled "government shutdowns"). So, to avoid violating this law, he simply cuts all spending across the board by the necessary amount to immediately balance the budget, everything. This would mean a technical default as interest payments on the debt would not be made. This has serious legality, but very unlikely.
3) Partial haircut. Avoid technical default by paying interest and principal on coming due debt, but cut other spending. This has many variations from applying (2) but not to the debt itself or also preserving some other categories not to cut, with pensions for veterans having perhaps the strongest constitutional argument for being preserved based on the specific language in Section 4 of Amendment 14 that speaks of pensions for Union soldiers in addition to the national debt as being inviolate. Some variatoin on this may be his most likely choice, legally problematic as it would be.
4) Mint high-value platinum coins. I have posted here on this idea of beowulf's, legal under a 1997 law. So, US Treasury mints trillion dollar platinum coin and deposits it with the NY Fed, continues to pay bills without having to borrow. This is indeed legal and would avoid a constitutional crisis, but would kick the can down the road on the broader debt ceiling and deficit issues, and would also probably be ridiculed and poorly received by the financial markets.
5) Have the Fed forgive portions of US debt it holds. This would allow for borrowing without breaching the debt ceiling, and is probably legal. However, no other central bank has ever done such a thing, as near as I can discern from some googling (although some have forgiven interest payments on debt), and would also be received poorly by financial markets. Also, House in particular would probably go after the Fed big time, led by Ron Paul. Indeed, I suspect that if Ben Bernanke and Tim Geithner were to discuss this, Ben would say to Tim, "you mint that coin."
I shall make one final note on the debt ceiling itself. Many are loudly declaring that it has always been there to "discipline" the budgetmakers, even though the budgetmakers are Congress itself and should tie the debt ceiling to their making of a budget, as I recommended in my most recent post here. However, back in 1917 when the ceiling was first adopted, it was done so as a mechanism to allow for flexibility on the part of the Treasury in connection with financing for WW I. Previously, in following the explicit mandates in the Constitution, Congress had always specifically approved (or disapproved) every specific act of borrowing money by the US government, much in the way one sees at state and local government levels. But the debt ceiling was put in place to allow the Treasury to engage in borrowing on its own, although within the limits set by the debt ceiling, very far from the current interpretations by so many people, including a lot of idiots in Washington who, as Paul Krugman describes them, claim to be Very Serious People.