If you have something plugged into the headphone port of the Mini, you can no longer choose the Mini's own speaker for output.
If you have two output devices, such as something plugged into the headphone port and some USB speaker, you can choose to have "Sound Effects" output through one output and everything else through the other. To do so, go to System Preferences->Sound->Sound Effects and choose where you want to direct your various dings and other alerts; then go to System Preferences->Sound->Output and choose where you want to direct everything else.
You could install something like SoundFlower and then direct the Sound Effects there, while leaving Output directed to the headphone jack. You will then not hear the Sound Effects at all.
As a final alternative, you can use System Preferences->Sound->Sound Effects to reduce the Sound Effects volume to zero without affecting the volume level for Output.