One more alternative option to DeltaMac's excellent advice, coming from a user of two Dell displays - one of which is "shared" between a rMBP via mDP>DP and a 2012 Mini Server via HDMI; I also own a small business with several Mini Servers connected similarly. I am not addressing that your Mini could have hardware problems. My offices house Eizo/Dell/BenQ displays with both PCs and Macs, and each configuration needs a bit of attention from time to time.
Most of the displays that are used in my house and offices are used with the same PCs and Macs every day. Switching a display's video source pretty much offers a similar situation to connecting your new/used Mini - it's a different source, and your Mini is also likely "used" to a different connected display. Supplementing DeltaMac's advice, I recommend resetting your display's settings (preferably) or unplugging your display for at least 30 seconds (at a minimum); once a reset has been performed, then connect your Mini and check your Display Pref Pane settings.
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