Recently my 2002 MCS has been making an intermittent noise/vibration when the AC is engaged. It happens every couple of minutes and lasts for about the same. It almost sounds like a bad bearing but I hear the compressors are pretty reliable. The AC takes several minutes to blow cool but it gets there. It is not nearly as cold as when I acquired the car 7 years ago.

Any suggestions on what to look at first? Coil, clutch, etc?

 

Compressor. Your mini is old enough that it probably needs replacing..

 

Might be the clutch on the AC compressor slipping. Metal spinning on metal sort of sound. They can be replaced, but often it shows the compressor is getting near its end.

The other thing to consider is the extra load from turning on the AC is stressing the belt tensioner pulley or idler pulley. You should be able to localize the sound if you listen carefully with the hood up.