I must be hopelessly jaded, but I am just failing to find Rep. Anthony Weiner's actions a resignable offense. I've been kind of fascinated by the topic of scandal-related resignations... well, for a long time... but really since reading Cramer's What It Takes, which chronicles some high quality presidential candidates (Gary Hart and Joe Biden) dropping out of the 1988 race because they were plagued by scandals (allegations of infidelity and plagiarism, respectively). What fascinated me in those cases was that the candidates didn't necessarily have to drop out. As far as we know, no one pulled an endorsement or threatened to stop funding them (although that may have happened if they'd stayed in the race), and the pullouts happened long before people started voting. They just became convinced that the horrible media coverage of them would never end, and that only by dropping out could they stop the assault on their lives and their careers.
Of course, had they stayed in the race, the horrible coverage would have ended, or at least changed focus in some way. Bill Clinton proved that. If you just take Matt Yglesias' advice and don't resign, the media will eventually find something else to discuss, or even find a different way to discuss you. Clinton, after all, went from the dope-smoking, draft-dodging womanizer to the Comeback Kid in just a few weeks.
Now, there are scandals and there are scandals, but given that Weiner, as far as I know, didn't engage in criminal activity and didn't send his tweets to a minor, I don't see why this is anything beyond bad judgment, which happens once in a while when middle-aged guys use social media. (Believe me, I know.) If Weiner finds the current media barrage intolerable, then sure, he should resign. But if he thinks he can handle it for another few days until the press finds something else to express moral outrage about, then he should stick it out.
Another option, as Jon Bernstein notes, is the Phil Gramm model: resign and then run in the following special election. It recognizes that there's a question over your conduct, and you submit yourself to the voters (who really have more of a say in this than Nancy Pelosi or Eric Cantor or I do) to see whether you should remain in office.
Would Weiner survive such an election? John Sides highlights an interesting piece of research from David Doherty, Conor Dowling, and Michael Miller showing how the public feels about various hypothetical scandals. Voters, it seems, don't like immoral actions, but they're more tolerant of those than they are of financial scandals or abuses of power. So, as scandals go, this one's probably relatively mild in the eyes of Weiner's constituents.
As for the question that so many news commentators are asking -- "How can Weiner survive?" -- the answer is pretty straightforward: by turning off his TV, and not reading (or posting on) Twitter for a few days.