DSP Hardware for Detecting an Audio "Click"
Probably the easiest (and most accurate) way to do this is the matched filter approach suggested above.  Basically, you can record each click independently and use those as matched filters against the microphone signal.  If the clicks are always the same, click A will ring up very strong on the A matched filter.  Same with B and C.  

To implement the matched filter, you can get a development kit with a processor or DSP chip on it.  It will take some coding to implement the solution (probably C).  In college, I worked on the "TMS320C6713 DSP Starter Kit (DSK)" but you can try Arduino or Beagleboard.  Your system will probably not be highly complex.  As for the microphone, you can get a relatively simple one that connects to the 3.5mm jack on the development kit.  (http://www.amazon.com/Pyle-Pro-PDMIC58-Professional-Handheld-Microphone/dp/B003GEBGA0/ref=sr_1_4?s=electronics&amp;ie=UTF8&amp;qid=1440204270&amp;sr=1-4&amp;keywords=microphone+3.5mm)

A matched filter works by multiplying the incoming signal by the expected stream of data.  

Example: 
If click A lasts 1mS and your sampling rate is 44.1kHz, you would have a 44 sample matched filter (the click may be much shorter).  You can store this in the code and point multiply your incoming signal by this matched filter.

MATLAB example code:
https://stackoverflow.com/a/19471009/3920284

The maximum value you get when you multiply by your input stream will correspond to the time when your click happened.  You can set a threshold like the example does.

One thing that may change over time is your clicks' sound: as the hardware ages, the clicks will sound different since wearing out may occur.