If it's an electrical contact sort of click, it may be a grounding issue. Basically, the strings are connected to electrical ground on purpose. As soon as your fingers touch the strings, you body then becomes grounded. With a normally grounded AC powered amp, this usually has the effect of turning your body into a large shield, and the hum level reduces. If you have ground noise, it's possible that when you touch the string you hear a click because you are basically switching the noise off (or on as the case may be in some situations).
If that is the case, you should ground your body by a more secure connection. Tin foil in your socks, connected to ground, or something like that. There was a Nashville trick of coiling some copper wire around your finger and connecting to ground (maybe an alligator clip to the strings behind the bridge which should be grounded). That will stop the noise being switched on and off as you touch the strings.
Otherwise it's probably mechanical noise ... light strings are worse. A good player on a good guitar with carefully selected strings is usually the solution for that.