Under this option, Obama would say he is ordering the Secretary of the Treasury to borrow funds to avoid a catastrophic global economic calamity only until such time as Congress passes a debt ceiling increase that he is willing to sign into law. This would not be a presidential power grab, but rather the exercise of presidential leadership in the national interest at a time when the political process has demonstrated an inability to function.
He could point out that Congress has passed a contradictory set of laws, ones that authorize spending far in excess of tax revenues, but prohibit the borrowing necessary to make up this gap. As president, he is choosing which of these sets of laws to follow until Congress takes action to reconcile them.
If Obama uses the Lincoln Option, he should emphasize that he is doing so openly, solely to avoid a national emergency, and only until such time as a new law is enacted. This would distinguish his action from the Bush Administration's reliance on inherent executive branch authority to take a number of counterterrorism actions in direct conflict with federal laws.
He should also emphasize that he would be taking this action in full recognition of the political risks it entails (a euphamism for impeachment). While I doubt that borrowing money in contravention of the statutory debt limit is a "high crime and misdemeanor," there are certainly many in the political sphere who will say it is.
But in the end, I think the people would be with Obama and even the current House of Representatives would be hard pressed to initiate an impeachment process when 14 million people are out of work and the President took temporary emergency action to save the economy.