There are two ways to leave residual radioactivity.
One is neutron activation, where atoms in a substance absorb enough neurons that they become unstable. Some atoms can absorb quite a few neutrons without becoming radioactive; they're useful for fission reactor control rods, as well as for shielding. Other atoms become radioactive by absorbing just one neutron. Technically, it's possible for particles other than neutrons to be absorbed and make atoms radioactive, but they must be much more energetic than neutrons, so all other particles add up to very little of what we might call "non-neutron activation". Anything left inside a fission reactor will gradually become radioactive due to neutron activation, because fission reactors are neutron-rich environments.
The other is fallout, where matter that is already radioactive is left behind. That happens from a dirty bomb, the fission products of a nuclear bomb, and to an extent, from neutron activation of some elements in soil in a nuclear ground burst.
Traveller doesn't tell us what kind of particles are used in its particle weapons. But the most plausible for a long range space weapon is neutralized protons. Either the weapon accelerates equal numbers of protons and electrons in some way that allows them to merge instead of de-focusing due to electrical charge repulsion, protons are accelerated and pass through a neutralizer stage that combines them with electrons on the way out the weapon, or the particles are accelerated by some hypothetical non-magnetic process.
By the way, proton accelerators are used in medicine. They have the useful property of depositing a gradually increasing amount of energy in the patient's tissues as depth increases, then a sudden drop-off beyond a certain point. That means that the energy hits mainly the target, with little irradiation to the surrounding tissues. That is similar to the description of the meson gun, but as it applies here, it doesn't leave meaningful irradiation behind.
