I suppose that you meant to say "mDP (Mini DisplayPort)", which is the other graphics port available on your 2011 mini.
Try SMC and NVRAM resets.
Do you get any error codes when you try the hardware test?
Have you tried a different HDMI cable?
Not a fix, but if the HDMI port doesn't work, after the above resets, and different cable - then use mDP to HDMI cable.
Here's what I do for an SMC reset, in addition to shutting off. I also have a mini.
Disconnect everything, particularly the power cord.
Press and hold the power button for about 5 seconds.
Plug in, turn on, with an NVRAM reset, let the boot chime 3 times before releasing the keys.
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.