Does the click occur when you go from OFF to RUN, or from RUN to START? If the latter, then the solenoid is duff, as it is obviously recieving switching power, but is not switching.
If there is no click at all when switching from RUN to START, then either the solenoid is duff (test for battery voltage on the trigger wire when at START), or there is a faulty interlock switch somewhere preventing the engne from cranking. Usual culprit is the brake switch.
If this is a model with electric deck lift etc, then either one or more fuses has blown, or the electronics have developed a fault. If this is the case, there is no alternative to replacement.
phil_saunders(a)bigfoot.com
Phil Saunders, June 2009