Why was an 8 TeV collider needed to find a 125 GeV Higgs? This might be very naive, but why wouldn't a (say) 209 GeV LEP do the job?
 A: You not only need enough kinetic energy to create the new particle, but enough to create the new particle after losing kinetic energy to all the shrapnel particles. At the very least, collisions produce a lot of photons that carry away energy due to the electromagnetic interactions between the colliding particles. Also, head-on collisions are extremely rare. Glancing collisions are much more common, so not all the energy in the beam goes into the collision. The original beam particles merely scatter off at glancing angles, taking away energy from the collision. Increasing the energy way beyond the amount necessary to produce a new particle increases the chance that even glancing collisions will have enough energy to produce the new particle.
In particle physics, it's not a matter of getting one collision to produce the new particle. Every collision produces a gigantic spray of particles that has to be sifted through for actually relevant data. We need to create the new particle many times to build up enough statistics to be confident that we're actually seeing something real. Higher energies allow us to produce more samples of a rare event to make it easier to study.
A: LHC is a hadron (proton) collider. But it's being used mainly as a gluon collider. Protons are composite particles, and at high energies they become a complete mess of quarks and gluons. While protons have a huge energy, the gluons that produce Higgs bosons only carry a small proportion of that energy. The rest of the energy goes to other gluons and quarks, that produce "undesired" jets of particles. The probability of each gluon/quark having a proportion of the energy is modelled by parton distribution functions.
LHC is a discovery machine. When it was designed, the Higgs mass was unknown. The fact that gluons get a variable proportion of the energy allowed them to probe a large range of masses at the same time. On the other hand, the next "big" accelerator most probably will collide electrons and positrons. Those are [believed to be] elementary particles, and can produce a Higgs boson directly, without any sharpnel particles. Therefore, the energy of the collision can be tuned to the Higgs mass. Such a machine would work as a Higgs factory, and would allow us to study the Higgs' properties in a more systematical way.
