Thanks, PhysicsGirl, for the thoughtful comments. I wholeheartedly agree that there is a lot planned at the LHC, much of which has nothing to do with the Higgs boson. And yet, the Higgs seems to be at the center of the popular discussion (as I pointed out, with the articles in the New York Times, the New Yorker, and the Boston Globe, in the last month) surrounding the LHC. The reason for this is that many physicists have identified the Higgs as good publicity: a new particle is easy to sink your teeth into; understanding mass generation is pretty easy for non-physicists to identify with. (It's worth noting, however, that although the Higgs gives mass to elementary particles, most mass that we observe directly comes from the potential energy between the quarks that make up hadrons, like protons and neutrons.)
The bad news for the LHC (and for particle physicists in general) has to do with this Higgs rhetoric. If a simple, vanilla, Standard Model Higgs is found, selling future accelerators will be difficult, if only because some new concept will need to be introduced. The cancellation of the Superconducting Super Collider in 1993 shows what can happen when an unsympathetic legislative body (or a public) can't be convinced that basic research is important. Coming up with something new (like further research into the quark gluon plasma, which isn't emphasized in most popular science accounts of the LHC) to convince taxpayers that accelerators are still useful won't be easy.
If something other than the simple, vanilla Higgs is found, then selling future projects is much easier: the theory was wrong, and all sorts of new things, in the same vein as the Higgs, may be found at accessible energies.
One point that I might clarify is that, in some sense, the Standard Model predicts when new physics will be found. Quantum field theories, of which the Standard Model is one, often have terms that go to infinity at finite energy scales. This means that new physics must occur at these scales. The minimal Standard Model, in which the Higgs behaves most simply, doesn't have any of these terms. In this case, it's described as "renormalizable," which means there's no primae facie reason to expect new physics. Of course, we know that near the Planck scale, gravity becomes strong, so something must happen there. But before that, it's difficult to imagine how new physics could enter the picture at all. This is why I conclude that, if the plain old Higgs is found, we're unlikely to find anything beyond it. And I think this would be bad news, since, as you rightly point out, there are many things unexplained by the Standard Model.
On the other hand, if some complicated Higgs-like mechanism is found, it's anyone's guess what the LHC will observe. In fact, many physicists have pointed to the things the SM does not explain as evidence that a simple Higgs mechanism is unlikely. And this may be. It doesn't bear, however, on the possible counterfactual that a simple Higgs mechanism gives little reason to hope for physics beyond the SM.
It's also worth noting that a number of physicists have pointed out that the details of the rumor may rule out a simple Higgs. This would be great.