Hook up 3 monitors to Mac Mini Late 2014
Apple's specs show that the late-2014 (which is still being sold today, October 2015) Mac Mini can support up to two displays using its HDMI and thunderbolt ports.
I've confirmed this by trying to plug in 3 monitors directly into the ports on the Mac Mini (two via thunderbolt, 1 via HDMI), with the same results as the OP...one of the monitors is not driven.
The following 3-monitor setup worked for me:

Late 2014 Mac Mini with 8GB RAM (purchased Oct 2015)
3 monitors, each with 1920x1200 resolution
Monitor 1: driven by Mac mini HDMI port (HDMI-to-HDMI cable)
Monitor 2: driven by Mac Mini thunderbolt port (I used this thunderbolt-to-DVI cable)
Monitor 3: driven via USB 3.0 adapter (I used this one from CableMatters and have had zero issues with flicker/performance, although I am not trying to play games or watch video on it...I use it for work).


Some notes:

You can use a different USB adapter (there are several on the market) but make sure that it's Mac compatible.  I previously tried one that had &quot;beta&quot; Mac support and it was terrible.
Thunderbolt-to-HDMI and Thunderbolt-to-DVI cables are really mini-displayport to DVI/HDMI cables....the Thunderbolt ports accept mini-displayport connectors.
For the two natively driven monitors (#1 and #2 above) you can use any combination of HDMI&lt;-&gt;HDMI/DVI or thunderbolt&lt;-&gt;HDMI/DVI cables depending on what your monitor accepts.