My mini cooper just makes a clicking noise when I try to start it

check for clutch travel and adjustment, as well as gear box oil level and correct grade, also make sure there isn't a floor matt hindering full pedal travel

If you try the window button and hear just a click from the inside of the door - the motor is fine! Problem is in the contacts of the switch that registers if the windows is all the way up. Try this method for quick fix from video below (I know crazy right? But this worked for me).
Long term fix - open the door, get the motor connector and use contact cleaning spray on it to restore connection. A lot of mini electrical issues come from corroded connectors

Gave the CV axle joints checked.

check for voltage going to the small solenoid wire on the starter. if you have 12 volts change the starter.

sounds like you are losing ground,make sure your battery terminals are clean and tight

if the clicking noise is from the starter,you can tap the starter while someone tries to start the car.if it turns over and starts then the starter motor will need to be replaced.if you can get to the starter,using the multi-meter you can check to make sure the solenoid is getting battery voltage at the wire that gets voltage when the key is turned.if you do and you have battery voltage at the larger wire on the starter then this will also indicate the stater motor has a bad spot in it and will have to be replaced.the clicking noise is usually an indication that the solenoid is working passing the voltage to the starter armature but the starter is drawing too much amperage to turn the motor over.either way its recommended to replace the complete unit as if the solenoid is bad the motor itself will usually go bad shortly after.

Did you get an answer to this? My 2004 mini cooper did the EXACT same thing this past Saturday July 18th 2010

bang inside door hard just above speaker works every time on my mini cooper 2003

With ignition switch on and engine off activate a window switch... listen for a clicking noise. if there is no noise then start checking fuses and relays. The fuse panel in on the left of the drivers legs. open it up it is well marked and in it you will find a handy fuse puller and spare fuses if they're needed. If you hear the clicking sound then try the Mini cooper Thump. Apply it to the doors midway between the large and small speakers (It may take two or three solid thumps) re-apply as needed

Could be the steering column. I had mine replaced under warranty for the same issue. This is a known problem for first generation MINI's. Talk to your dealer for a replacement.