As Matt Yglesias suggests, one of the big reasons for the left bringing up the "Constitutional option," in which the president would override the debt ceiling by invoking the Fourteenth Amendment, is as a bargaining position: it gives the White House the opportunity to say to Republican negotiators that it doesn't need them and has other options if they refuse to vote to raise the debt ceiling.
It's pretty clear why the "Constitutional option" could be good partisan politics for Democrats. If the debt ceiling stands and government budgets run into it, we could enter a period of intense economic turmoil. That turmoil could sink the president's chances of reelection (that is, if he doesn't successfully deflect blame to the Republicans for the debt-ceiling fiasco). By giving the president leverage, this "option" can also allow him to parry cuts that he or his allies find unfavorable.
However, the "Constitutional option" may also be a partisan opportunity for Congressional Republicans. One of the many dirty little secrets of the current debt-ceiling talks is the fact that many Capitol Hill Republicans are quite glad to have trillions more in national debt. The GOP has voted for budgetary measures that would bust the debt ceiling. Moreover, the Ryan budget, backed by the overwhelming majority of Congressional Republicans, adds trillions to the debt and places most of the savings (some of which are premised on fantastic economic numbers) many years down the road.
But while Republican members of Congress have de facto embraced more debt, they also face a grassroots element and certain self-anointed tribunes of the conservative "movement" who increasingly believe that any vote to increase the debt ceiling is a sign of fiscal capitulation. Many Republicans thus find themselves between the rock of fiscal reality and the hard place of political positioning.
The "Constitutional option" offers the GOP an escape hatch. If Obama used this option, the GOP could keep on voting for massive deficit spending while also not making the politically hard vote about the debt ceiling. Obama's exercise of the "Constitutional option" would also give Republicans an opportunity to rail against an "out of control" executive branch while not forcing them to do anything to rein in this executive.
The leaders of both parties certainly want the debt ceiling to be raised, but they are trying to find the least politically damaging way to do it. Republicans and Democrats may find that the "Constitutional option" is the easiest way of doing so.
Of course, the use of this "option" would very likely be a ticket to a Constitutional nightmareland and could set up a further corroding of our federal institutions and national consensus. The use of this option may be good partisanship for both parties, but it seems bad politics for the nation as a whole. It would be far better for Congressional leaders to hammer out some compromise that would allow the United States to meet its fiscal obligations over the short and long terms.