What is the best mini PC for programming?

The best mini PC for programming is one that ticks all the boxes you care about. Here are some points to consider:

Twice the memory working set so it doesn’t have to swap, and has some over for buffering (I have 32GB) — allow for operating systems and applications to double in size each 3 years;

 SSD for primary disk, HDD for archive / backup;

Best CPU you can afford — if you spend more than 1 second waiting for the computer to respond. Basically you want enough CPU power and cores so it can run your IDE and two copies of the application (development example and auto testing in background);

Enough displays so one display can be your IDE, one can be what the customer normally sees, and optionally a third for reference material; 

[I run two HD screens off the laptop]

Portable if you work in multiple places [I picked the Toshiba Satellite P870 because I wanted 32GB of memory, two disks, and a 17″ screen about the same pixels per inch as the screens I was going to also connect]

A decent keyboard and mouse — I personally use a Logitech K800 keyboard and Performance MX mouse, and take at least the mouse, sometimes the keyboard when I go onsite. Pick what is your own personal preference, and if necessary adjust the sensitivity of the mouse.

 Pick the operating system that you like best for the work you are doing (I prefer Ubuntu LTS linux, and run windows in virtual machines as needed);

[Edited to add peripherals and OpSys at Alex’s suggestion]

The only downsides of having a system that is more powerful that some production systems, are a) that you are much less aware of any accidental inefficiencies and b) with 32Gb of memory my system takes longer to come out of hibernation than it does to boot cold (on linux) because hibernation is a full memory copy, whilst boot isn’t.