Turn the Mac Mini upside down so that the drive can be seated more easily (with the help of gravity) and it should lock into place. Worked for me after a little fiddling.
This also worked for me. Follow machobbes advice - whole process took about 10 minutes. Easy peasy.
The one difficulty I had was getting the new SSD (or HDD) to lock back into the fixings inside of the casing. Best solution if you are having this problem is to turn the Mini upside down, ensuring that the antenna cable isn't stretched too far, and aided by the wonderful powers of gravity, get the new HDD unit fixed into place properly.