Mac Mini making loud noise

A video stream on pause is not quite as inactive as it might seem, because the system has to service the application, the network connection and thus creates CPU load and system activity even if the video is on hold.

As such, it would not be entirely surprising that you might have some symptoms of a busy system under those circumstances. It's also entirely possible that it might impact an external drive even if that is not directly involved in either the application or data.

To test out whether the external is the cause or not, simply dismount it and power it down. Clearly if you still hear the noise, then the external is not at fault, while if you don't, then external is a distinct possibility.

Could you tell us what else is going on when you hear this noise?

For example, are you using the superdrive at that time?

Or playing a video clip?

The fan isn't likely to make a grinding noise, though the superdrive might, if it were playing a disc.

There is nothing that should cause such a noise if the system is otherwise inactive, unless there is a CD or DVD in the optical drive which the system briefly spins up - and even that would depend on what is on the disk.

Since it can be rather difficult to tell whether this kind of noise is the optical drive or fan, since they can make similar sounds, I would first see if the system makes this noise at any point after being booted when there has not been a disk in the optical drive at all since the startup. If no disk has been inserted, the optical drive should not have any reason to spin up. If the noise still occurs, then download a copy of MenuMeters ( http://www.ragingmenace.com/software/menumeters/) set it to display all counters including the one for paging, and then monitor the system. As and when the sound recurs, check if there is any sign of high CPU or prolonged drive activity at the same time. If so, then the sound is likely the fan being spun up to deal with load on the system, and it's a question then of finding what it is that is causing the activity, by using Activity Monitor (in Applications>Utilities). If there is no high CPU or disk usage, then the sound is most likely a fault condition, and if the mini is the unit outlined in your profile (ie, a core 2 duo 2.0HGz model) then if it were mine, I would contact Apple to discuss the problem, or take the system to an Apple store or service provider to have it looked at.