I’d love a display that let me pick the color or picture showing on the screen when it is off, or maybe it’s the wrong question and power-efficient things with solar panels directing sun energy to batteries that keep them going on all the time is the way.
To make them go white, they actually have to use power to clear the image. If you just let it sit there and let the battery die, the image will be there until you recharge it and turn it back on.
E ink is really cool tech (just patented and expensive). But the daylight computer from the other comment might be more what you're looking for.
And yeah you can totally run a solar setup. Anker makes a bunch, for example, of portable li ion batteries that can be directly connected to their solar panels.
Edit: If what you want is a digital billboard or picture frame, there might be some of those out there too, or I think I saw some DIY projects building those out of eink panels and an Arduino or such. Some of them get commercialized on Kickstarter.
I’m excited for the future with solar, battery, and really most electronic technology becoming cheaper and simpler.