with three devices plugged into it.
I have found that for the HDMI switch to recognize another device, automatically, the other devices have to be turned off.
I got around that by using a HDMI switch with its own remote control which was soon replaced with one of those Harmony Remotes.
You didn't specify how many ports your switch has, or whether it has its own power supply, but you may find that one of those ports is the master.
HDMI should supply 55mA on the 5v supply so, theoretically, your device should be getting enough, as long as the mini meets HDMI specs
I recently purchased a HDMI switch (Just Hook It Up 2 Way HDMI Switch) to connect my Mac Mini (2012) and my custom desktop PC to a single monitor. When the PC is turned on, the switch works correctly and I'm able to use my desktop. When I turned on the Mac Mini, the switch doesn't seem to recognize that I actually turned the Maci Mini on and the monitor just shows the typical "no signal." I'm wondering, since most likely the Mac Mini draws a lot less power than the desktop PC, is it possible that there isn't "enough power" for the switch? Should I find a switch with its own power source, or is there something unique about the Mac mini that doesn't work well with HDMI switches?