Aha! I see someone here has been listening to too much Pink Floyd!

But ya: the "Dark Side of the Moon" idea is just metaphoric. As others have pointed out above, that side of the moon actually gets plenty of bright sunshine.

BUT... just for the fun of it...

Let's say that somewhere out there in the universe is a world where one side is always in darkness. In fact we've already actually found lots of those worlds! There are many exoplanets that have one side of the planet "tidally locked" to its star, so that one side is always in perpetual day-light, while the other side is ALWAYS in perpetual darkness. For ever and ever.

So to answer your question for a probe visiting a world like that...

And wanting to image the dark-side of the planet (which is literally dark)...

Then how would they do it?

Well, you're on the right track: bringing a bright light is certainly one way of doing it!

Except that instead of having your light being bright in the visible wavelengths, it would be bright at microwave and radar frequencies.

So, in other words, we could simply just scan the surface with a bright (powerful) onboard radar system.

In fact we already do this! We often scan celestial bodies in our solar system like this, using radar. For example, sometimes when a small meteor passes close to Earth, military radar facilities in the USA will scan the meteor with powerful radar beams, just to give us a general idea of the shape and surface structure.

If I'm not mistaken I think the giant Arecibo Telescope in Puerto Rico also scans objects in the radar frequencies.

As well, we've scanned Venus with radar, and other worlds including Saturn's giant moon Titan.

In addition... there's another way of doing it as well, that would NOT involve bringing a "bright light" (aka: radar).

That alternative method is to simply bring a highly sensitive camera.

So instead of a bright light, you do the opposite, and bring a camera that can see really well in the dark.

There are cameras sensitive enough to take great pictures just using star-shine, and even the light emitted by the center of our Milky Way Galaxy. So basically the stars have just enough light on the night side, to allow a sensitive camera to use that starlight as a kind of camera light.

Using this method, you'd probably also want a long exposure time to really bring out the details even more, so you'd use photographic tools, such as taking repeated photos of the same spot, on each orbital pass, and multiply/stack those photos, along with image processing statistical analysis/enhancement, etc...