Please read some more forums on Honda Civic 2005 starter problems - you might be able to pin point the fault.

http://troubleshootmyvehicle.com/honda/1.7L/starter-motor-diagnostic-test-1

A rebuilt starter is not a bad idea for a 10 year old car.. just check the reviews for quality and performance.

I found internet forums to be more helpful than the mechanics. 2 years ago I used to have a similar problem with my minivan - it would randomly not start. The key will turn, but not click, no start nothing. 10 minutes later when CAA arrived to tow the van, it started without any hesitation - just to make me look like a fool.

Eventually with help from internet forums I discovered that the production batch of the model I have was fitted with a failing transmitter which would stop the communication between the immobilizer and the car's computer. $150.00 later had a new module fitted with a new software download and the problem got solved. Prior to the forums, the dealer wanted to charge $99.00 for diagnostics and another $250.00 for the estimated repairs. Armed with information from the forums, I negotiated with a few dealers and got the job done for $150.00. If I had faced this problem as couple of years earlier - it would have been done Free under warranty. All this despite the dealers having a TSB notifying of this issue since 2006.

Just my 2 cents.

Hiren