Could be solenoid, starter or the engine ground connector. Is the click definitely from the region of the starter under the hood
Try turning on main lights so you can see them against a wall and then the starter.
possibilities
1) lights go out someone else will have to answer: Does the civic have a power saver relay that turns off major consumers when you try to start? and, are the main ligths included in that relay.
2) lights go very dim but not out batt voltage is dropping severely. shorted starter, shorted solenoid OR bad engine ground/battery ground. Could be that the starter drive isn't working but then I expect you would here the starter motor running.
3) lights stay bright starter solenoid could be bad or the main heavy current connection between batt and starter is disconnected and the low current relay connection to do the solenoid is working.
and many other things possible but the solenoid does click so there is voltage getting to that.
I am not familiar with civics but I'd wiggle the thick wires batt to starter and batt to body and see if they are loose or corroded in any obvious ay.
There is a currently a Jim Kerr article on this on the CD website
(Message edited by tpl on June 17, 2005)