What is needed to claim the discovery of the Higgs boson?

NOTE: I recommend reading Noldorin's answer first, for useful background information, and Matt's answer afterward if you want additional detail

Noldorin is right that there isn't a single event that you can look at and identify a Higgs boson. In fact, unless the theories are drastically wrong, the Higgs particle is unstable and it has an exceedingly short lifetime - so short that it won't even make it out of the empty space inside the detector! Even at the speed of light, it can only travel a microscopic distance before it decays into other particles. (If I can find some numeric predictions I'll edit that information in.) So we won't be able to detect a Higgs boson directly.

What scientists will be looking for are particular patterns of known particles that are signatures of Higgs decay. For example, the standard model predicts that a Higgs boson could decay into two Z bosons, which in turn decay into a muon and antimuon each. So if physicists see that a particular collision produces two muons and two antimuons, among other particles, there's a chance that somewhere in the mess of particles produced in that collision, there was a Higgs boson. This is just one example, of course; there are many other sets of particles that the Higgs could decay into, and the large detectors at the LHC are designed to look for all of them.

Of course, Higgs decay is not the only thing that could produce two muon-antimuon pairs, and the same is true for other possible decay products. So just seeing the expected decay products is not a sure sign of a Higgs detection. The real evidence is going to come from the results of many collisions (billions or trillions), accumulated over time.

For each possible set of decay products, you can plot the fraction of collisions in which those decay products are produced (or rather, the scattering cross section, a related quantity) against the total energy of the particles coming into the collision. If the Higgs is real, you'll see a spike, called a resonance, in the graph at the energy corresponding to the mass of the Higgs particle. It'll look something like this plot, which was produced for the Z boson (which has a mass of only 91 GeV):

The image is from http://blogs.uslhc.us/the-z-boson-and-resonances, which is actually a pretty good read.

Anyway, to sum up: the main signature of the Higgs boson, like other unstable particles, will be this resonance peak that appears in a graph produced by aggregating data from many billions or trillions of collisions. Hopefully this makes it a bit clearer why there's going to be a lot of detailed analysis involved before we get any clear detection or non-detection of the Higgs particle.

David Zaslavsky's answer goes most of the way toward answering the question "what is needed to claim the discovery of a new particle?", but to claim discovery of the Higgs boson, one needs much more. The first sign that the particle really is the predicted Higgs will be the decay modes it shows up in. The Standard Model makes a specific prediction for the rate of different types of Higgs events as a function of its mass. It could decay to two Z bosons, each decaying to two muons, as David Zaslavsky explained, but only if it is heavy enough. If it's lighter, it would decay mostly to bottom and anti-bottom quarks, which are quite difficult to see because ordinary strong interactions produce them much more often. It would also decay rarely to two photons, but this is a distinctive enough signal that it's relatively easy to study. The rate and type of events that are seen would be checked against predictions of the Standard Model for the observed mass. For instance, if a signal showed up involving 4 muons, but at a mass of only 100 GeV, we would know that we weren't seeing the Higgs, but something stranger and not predicted. Or if a 160 GeV particle were found to frequently decay to two photons, we would know that it isn't the Higgs, which at that mass would decay mostly to a pair of Z or W bosons. So there are a number of consistency checks to make between the mass and decay modes. Still, much more is needed to really say that it is the Higgs. For one thing, it should be a scalar, i.e. a particle without spin. The spin can be tested by looking at the angular separation among decay products. The Higgs also interacts with known particles in very specific ways; as many tests of this as possible would be performed. Even if the particle is not the Standard Model Higgs, it still might be a type of Higgs boson in an extended theory, such as the supersymmetric Standard Model. The road from discovering some new particle to having a completely convincing theoretical account of what the particle is could potentially be a long one, and might even involve other colliders in the future (like the proposed International Linear Collider).

For now, of course, we're all just hoping for something to be discovered; once this happens, the work of pinning down precisely what the discovery means will be exciting and will continue for years.

I'm not a particle physicist, but here's a little overview from what I understand.

Colliding particles, in particular hadrons (being composed of three quarks and a field of gluons), are quite capable of generating very "messy" collisions, the more so at higher energies. By "messy" I mean that the variations on the possible results of the collision (the number of different Feynmann diagrams) is pretty large. Of course, certain outcomes have much higher outcomes than others, and the probabilities of such can be estimated in quantum field theory. In any case, there can be decays into all sorts of fundamental particles (with different charges, spins, masses, etc.), and then further decays, and so forth.

Verifying that a particle (here the Higgs boson) is really what the theory predicts takes lots of runs of the experiment, and is largely just a complex probability game. Given that the properties of the Higgs aren't exactly known, what particle physicists are looking for is signs of a missing particle. i.e. A violation of some conservation law in the collision (typically energy or momentum). This is a good hint that there's an unknown particle that hasn't been accounted for. (For example, neutrinos are far too weakly interacting to be detected directly, and were first discovered by noticing some small energy was missing).

At the moment, all we really know from previous experimental data is that the lower bound for the Higgs boson mass is 115 GeV/c² - pretty high, but theoretically well within the range of the LHC. This helps us a bit in knowing where to look, but at the end of the day, it's smashing together protons over and over again that rather crudely gives you the final discovery!