OK, onto DisplayPort. I wondered what the difference was between this and HDMI, and couldn't really find a clear answer, so I ordered a cable to try for myself. Some say that it helps with monitors that run at 144hz, but I have no idea if it really does anything for a 60hz monitor.
I plugged the cable in and right away noticed that the picture was brighter. The image quality also seems a little sharper. OK, cool.
I woke up this morning and got no signal on the monitor. Plugged HDMI back in and got a signal. Did some testing this evening:
-This isn't a problem with the cable.
-It's not a problem with the (multiple) displayport inputs on the PC.
-Changed the nVidia power settings to maximum, based on a recommendation. No effect.
-I had a mini displayport-to-displayport cable for my Surface. Got no signal with it too.
-Unplugged the monitor (including the power cord from the outlet) for a few minutes, based on a recommendation I got that didn't seem like it would work...but it did.
-Plugged the monitor back in. Got a signal with the Surface.
-Put the Surface in standby. Got a signal but it reset the display settings I had selected.
-Plugged the displayport cable back into the PC. Got no signal.
-Unplugged the HDMI cable from the PC and got a signal.

Am wondering if this is just too much 'power' for the monitor to handle. Maybe it supports an earlier standard of displayport or something? If it happens again, it's back to HDMI. Just weird.
EDIT: The cable was from CableMatters. I see people complaining about their quality. Back it goes-- made sure to check displayport.org and went with a VESA-certified one instead.