What process does the LHC use to make new isotopes?

The main purpose of the LHC is to generate fundamental particles like the top quark and the Higgs boson. Other facilities, GSI and FRIB (under construction) being two examples, are focused on the generation of new isotopes and elements. But sticking to the LHC...

The proton-proton collisions at the LHC do produce energetic particles that will interact with detector components and will alter their isotopic and elemental content. In some cases this is done intentionally to measure the particle energy in calorimeters. In other cases the energetic particles produce undesirable radiation damage which leads to increased electronic noise, weakened metals, and clouded plastics.

The calorimetry measurement process is the most similar to a cosmic ray. As particles enter the calorimeters they will interact with a dense material (lead is a favorite) and produce a shower of new particles (in hadronic showers these are spallation out of the struck nucleus). The shower of particles then passes through a plastic, liquid, or crystal which produces light that can be measured. In a cosmic ray the calorimeter is the atmosphere and the shower is detected on the ground via light output in some of the same materials.