it does sound like a lowvoltage/bad connection problem between battery solenoid and starter but this is similar to a problem i had once.

this is a long shot, because your symptoms are not exactly like mine. and your car is not exactly like mine( this was ina 1991 integra), but MAY use the same part. this could be the beginning of what i had

i called acura service about it and they knew what it was right away. my car began giving me fits, i would have to turn the key several times to get it to start.(i THINK it was clicking) over the months the amount of time it took to get started never changed, but occassionaly it wouldnt start at all and would leave me stranded up to an hour or more turning the ignition on and off on and off on and off.....

it drove me nuts, i couldnt find the problem anywhere, then the car started just shutting down occassionally. WHILE DRIVING!

SOLUTION: called acura, said it was a common problem. All i had to do was buy and install the $80 ignition wiring harness. it runs from behind the key switch and to the fuse box. it apparrently starts losing a connection in the start position and eventually loses connection in the run position. i believe that in the beginning it had just enough contact to make a "click" but with this bad connection i would sometimes smell melting wire insulation.

HINT: if this is your problem and your car eventually dies on you(shuts off completely) you can jiggle and hold the key in the run position untill you get to a parking place.(this is a bitch in a manual though.)

i have no idea how to test this, and dont know what to suggest. i am just writing in case my symptoms sound like yours.

it is not a hard part to replace, but it isnt that fun to dissassemble and especially re-assemble the steering column cover just to poke around.