If Janet Napolitano resigns, then the Obama administration will be in deeper trouble than it already is.
As bad as the total failure of the national security apparatus to stop the Christmas terror attack looks now, it will look even worse if the secretary of homeland security resigns as more are calling for her to do. (The terror suspect didn't "attempt" to attack the plane, but did attack it and failed to murder.) There are a collection of reasons why Napolitano leaving would be worse than her staying for the administration.
First, Napolitano's resignation would remove any deniability for the failure to stop the attacker. At least some of the blame deserves to be put on security in the Netherlands and Nigeria for failing to catch the suspect's explosives. Second, Napolitano is in hot water not because of a personal scandal (except maybe saying "The system worked") but rather because the department she runs failed to recognize the suspect as a threat and stop him from being allowed to fly to the U.S. If the scandal were personal, Napolitano could leave government and take the scandal with her.
The true scandal is that the federal government seems unable to protect us from being killed en masse by methods similar to those used eight years ago on September 11 that killed more people in one day than have been murdered on American soil since the Battle of Antietam in 1862. Airline passengers -- not U.S. Marshals, not the TSA, not the CIA -- have thrice now prevented airline attacks in progress from being bloodier than intended: Flight 93, the Shoe Bomber's attempt, and last week's attack over Detroit.
Napolitano probably won't be fired, because she is a cabinet official, which is important for two political reasons. If she goes, it would be the highest-profile resignation in the administration after a year of significant departures, including the White House counsel, the communications director, and "green jobs czar" Van Jones. More importantly, the administration would enter a weeks- or months-long confirmation melee in the Senate that will give Republicans free media to savage Democrats on national security at the start of an election year. Democrats won't want to kick off the midterms by getting in a national security debate that has often cut against them in recent years.
The White House likely realizes all of this and will do with Napolitano what it does with Joe Biden after his gaffes by hiding her. Napolitano's home state of Arizona would be a nice escape from D.C.'s blizzards (meteorological or partisan). Then again, if she wishes to stay closer to the capital, Dick Cheney's "undisclosed location" is available.