I am not in front of the computer at the moment, however the monitor did turn on, and had power.  Is that what you meant?

I have a desktop that I leave running all the time. The monitor (connected by HDMI) goes into power save. A key press or mouse flick turns it on.

If I shut the monitor off for an extended period (a week), the monitor will not wake up. I have not figured out why. I have to unplug the monitor, wait, plug it back in and then turn it on. Then it works and stays working.

That is what I suggest you try

If You have HDMI connections probably you have also button to switch input.
Maybe you should use this SOURCE selector button to check if will work.
Also for your video card probably there is a control panel software. Make sure when you connect second monitor that Multi display option is enabled.

I set a profile on my monitor, checked the settings (1 monitor for me), saved the settings and turned it off. It is off right now.
.....................   Wait some minutes.
.....................   Over 5 minutes off.
.....................   Power button

AND,  The monitor will not start.  I have to unplug it, wait and plug it back in.

Once running, I leave it ON, I can turn the machine OFF and leave the Monitor in Power Save and it will unfailingly start up after any time off (weeks).

However, if the monitor is turned off, I must reset it by unplugging it and plugging it back in.

@ Tom..  thanks, do you mean a switch on the computer itself?  The monitor only has 2 HDMI connections only, and the computer only has 2 Display ports and 1 HDMI ouput..

What is the brand and model of the monitor?

In my case (and I know you are asking the author), my Monitor is a ViewSonic 2400 series Touch Screen Monitor

No, I was thinking about button build in to your monitor, and If you don't have it there is an option to switch INPUT going to Monitor setup.
It should be button to go to setup on sight or underneath your monitor

HDMI monitors (or TVs) require a handshake with the computer to provide correct signals/resolution and port selection.  So, if the monitor is powered off, when the PC is turned on, the hdmi handshake is missed.  

Some monitors, even if powered on, will not handshake unless the computer's hdmi cable is connected to the currently selected port.      

So, if the PC doesn't get the handshake on the hdmi port, it probably goes to its default.  In your case (assuming the hardware and cables are all good),  I would guess the default is the first Display port.   You might be able to adjust that in the controller settings or the bios.  

The recipe should be, power on the monitor, make sure the hdmi port where the PC is connected is selected,  then power on the computer.

Once the handshake is made, they should remain good, even if the PC or monitor sleeps.  If you have trouble recovering from sleep, then maybe lower the PC sleep level in the bios (S1, S3, etc.).    If you power down or reset the PC, it should handshake, even if the monitor is sleeping -- but not if its powered off, or in some cases,  selected to a different port.

This morning I plugged the monitor back in and it worked, so I don't know why it worked, however thankf for your suggestions.

Thank you. HDMI connections can be picky so your above steps are what you have to do sometimes.