How do I make the Mini Cooper (model 10242) motorized?
Here is the only example I can think of.

We can see an example in the following video @0:39 by PPUNG DADDY:

https://www.youtube.com/watch?v=kvVJXhs_Q3s&amp;t=38s

Here is the servo-motor:



And the IR receiver on the rear, next to one motor. The battery is located behind the driver's seat:



We have a better view at the end of the video: 
https://www.youtube.com/watch?v=kvVJXhs_Q3s&amp;t=1m31s



In order to do the same build, you would need:

What we clearly see in the video:


Rechargeable Battery Box (or the cheaper but bigger Battery Box)
Servo-Motor
IR Receiver
L-Motor


What is not shown but necessary:


a remote controller: IR Remote Control or IR Speed Remote Control (allow to control speed of the motors but it responds more slowly)
mount front wheels on a steering: see the front of this car for example: there are some fixed and free parts
mount rear wheels on rotating axis with a differential gear between the 2 wheels
some gears in order to transfer the movements from motors to the wheels:


direction on the front wheels from the Servo-Motor (but there may be some simpler setups)
gears between the differential and the L-Motor



These pieces would cost between 80 to 100 $, € or £ (I used the Brickset's average prices).



I don't have this model so I can't tell you what exact parts are required, sorry.