I agree with Vasco, Mini PCs are mainly used for common tasks - writing text, spreadsheets, and presentations in office software, surfing the internet, watching videos, viewing photos, and accessing Web tools and applications. Buy the time you get one powerful enough to handle games you really aren't saving anything and you would most likely have to lower the down the graphics detail and resolution to 1366x768 to get decent frame rates out of it. The Intel Iris Pro Graphics or Iris Plus will be able to do some graphics.
If you are looking to get a mini for some reason then why not get one with a Iris graphics in it so that you can do some light gaming. If your primary purpose with be to game your money would be better spent on a laptop or desktop that can be built or order to your need. For instance, the Intel Iris Pro Graphics 580 NUC costs over $600 and won't include a monitor and only has a single HDMI port. For the same price you can get a budget gaming computer with a RX 550 in it that you can remove later on when you need more power and replace it with a better card and the RX 550 is over twice as fast as the Intel Graphics. The computer also comes with a keyboard and mouse.
For $200 more you can get a system with a GTX 1060 and now you a real gaming computer that can play most modern games at full 1080p resolution and high details. Or how about The Dell Inspiron 15 7000 Gaming laptop that has a GTX 1060 for around $900? It would be inadvisable, in my opinion, to purchase a mini computer to use as a primary gaming device.
Considering ED is best experienced in Native 4k at 3840x2160, I would say there is little motivation to bother.
Well, there you have it... Tote that on your train!