The current Mac mini is to put it bluntly - obsolete. It is as you stated a 2014 model meaning it was designed more than four years ago which in turn means that in computing ages it is (almost) prehistoric. Even when brand new it had a rather feeble video chip and now it is utterly pathetic.
In a partial defence of the Mac mini - it was intended to be the cheapest lowest end of the Mac product line.
This means it has very limited capabilities in terms of resolution, it can in theory do 4K but it is so limited it cannot properly do 4K at that standard 60Hz refresh rate. As a result you can buy many, many good cheap monitors that the Mac mini is simply not capable of properly supporting.
As is almost always the case the best results are from using the Displayport connection rather than the HDMI connection.
Because the Mac mini is borderline at achieving even 4K it is hard to suggest a definitive monitor, I would therefore advise only buying if you can test before paying or can return if it does not work. This leaves three options.
- Get a standard 1920x1080 display, these will work no problem
- Get a Displayport 4K display and try it
- Get an ultra wide display like the LG ones, these are wider than normal but the same resolution vertically e.g. 2560x1080 again you should choose one that uses Displayport as this definitely exceeds the Mac mini HDMI capabilities
Arguably for what you intend an iMac 27" which has a resolution of 5120‑by‑2880 would be the best choice, it would however cost more. Consider selling your Mac mini system to raise money towards the iMac.