Not done with you yet, OP! Before you investigate hardware issues (but don't stop budgeting for a new Mini yet!). There's two SW tasks yet to try.
I have posted here, last year sometime, that it appears that Apple started "cracking down" on display connections and compliance. With compliant display cables, my Dell and BenQ displays have more built-in features today than when I bought them. Ditto my HDMI-connected displays.
On the above note, there's one more bit you should attend to IMHO that might be gumming up your works - also mentioned by me in these forums when I discovered what was happening with my non-compliant DP cables and started using SwitchResX before I educated myself - I own/run a small company with 50+ engineers and you'd be surprised how whiny they can be when things don't work right. My strong recommendation to you here is to delete the DisplayID override data associated with your displays - these are the profiles that tell your Mac what your display's settings are; look up EDID for a bit of background, DisplayID is EDID's evolution. There's an "override" for each of your displays relative to each port their connected to - a profile is created for a display connected via HDMI and another profile is created for that same display connected via mDP. The DisplayID profiles are located in the
/System/Library/Displays/Contents/Resources/Overrides/ directory, with many defaults set up by Apple. OS X/macOS creates a new folder within the Overrides directory for each vendor not in Apple's "list" - for instance, my Dell displays settings are in a "DisplayVendorID-10ac" folder, with profiles for each of the Dell displays I connect to. I deleted the entire contents of the Overrides directory - with my displays connected, then restarted my Mac - OS X/macOS rebuilds the relevant contents of the Overrides directory. You'll literally start with a clean slate…
In your OP you indicated that neither device connected via HDMI are seeing each other. Have you tried the "Detect Displays" command, which has been hidden by Apple since Mavericks. Connect the Mini and displays as you normally would, I always restart the Mac at this point, open the Displays Pref Pane, hold down the Option Key, and the Detect Displays command button appears. This command sometimes fixes display issues as you're describing. Nope, I have no idea why Apple tucked this command away…
Option (no pun intended!) two is a bit more in-depth but easy to accomplish. I move to my 3 offices regularly with my rMBP in tow (my Mini Server stays at my home office, but it's connected to one of my two displays via HDMI while my rMBP is connected to that same display via mDP and another display via mDP). The routine - provided to me a couple of years ago by Apple Support:
Power down the Mac
Disconnect the HDMI cable from the Mac and display and wait about 10 seconds
Reconnect the HDMI cable to both devices
Using a direct-connected keyboard (BT keyboards don't work here), while holding down the Option Key, simultaneously power on the Mac AND press Command+F1 (I think this keyboard combination was first used with the Ti Macbook Pro (I used it on my 17" laptop to toggle video display modes for external displays between extended and mirroring modes).
Optional: Depending on how you connect to your network, you may see a icon of your hard drive and a dropdown asking for you to choose your wifi network (I connect via ethernet most of the time, so this isn't an issue for me. If you do not see this startup screen, start the process over again.
Your second screen should now be detected. Not done yet. Restart your Mac.
[doublepost=1501536151][/doublepost]Also, besides replacing the logic board, the two other options are checking out multiple display solutions by Matrox and signing up to be a Developer and buying one of their nifty eGPUs (which I'm considering for my Mini!). Cheers!