My '13 LT 6spd manual has done this three times. First time I turned the car off with the A/C on. Did what others did and turned key to ON to check idiot lights, saw nothing, then turned the car OFF, and it stopped.
Second time it happened, I was at my hackerspace. Rushed in and grabbed a friend that was there who might be able to pinpoint the location. Couldn't, as the sound slowly faded away.
The third time it happened, I had driven about 45 minutes, but only had the air on about 20 minutes into the trip and turned it off about 10 minutes before arriving. (I only had the air on while driving on the expressway). When I arrived at my destination with the air turned off, the car made the noise. I did the key trick to turn it off. Not sure if that's bad or not.
To the person in post #68 (too lazy to find it and quote it), the reason why it doesn't stay on due to the key turning to position ON then OFF, I'm guessing based on my BS in computer engineering, is that the way they designed the system (if this is normal), is that the ECU looks for a value that is greater (or less than) another, and then turns this solenoid, motor, etc ON for X amount of time (perhaps based on the initial read value). When you turn your key ON then OFF (interrupting the current operation), the ECU reads again, but at this point, the data value it's checking is within the safe threshold, so the solenoid, motor, etc. does not kick back on or stay on. Just my two cents.
The way to prove me wrong is to be able to never get it to come back on after you go back ON to back OFF once you turn your car off, immediately after hearing the sound.
The turbo lube idea was pretty good. But since people with LS Cruzes are saying they hear it to (granted they're hearing the same issue), then it leads me to also believe it is the A/C unit. I have never heard it through the winter, even with defrost on. Only on hot days (I rarely use A/C).