Mercurial Base

A Mercurial Base is a large installation on a planet extremely close to a star—so close that direct sunlight would be lethal. As a result, the base is continually moving, to stay in either perpetual twilight or perpetual night.

More plausible versions (relatively speaking—this isn't a particularly plausible idea) often put the Mercurial Base on train tracks, and use the thermal expansion and contraction of the tracks to move the base; this only works at sunrise or sunset, which can result in added drama. Less plausible versions might move it in just about any way the author thinks is cool, and are probably more likely to try to stay near midnight, or just after sunset, to give themselves a larger safety margin.

Because of the Rule of Drama, a Mercurial Base inevitably comes along with one of two events. Either:

some major character will get stranded on the surface, and needs to get back to the base before the sun rises, or
the base itself will break down.

At least as applied to Mercury itself, the Mercurial Base is of necessity a relatively new idea. Astronomers used to think Mercury was tidally locked to the sun, with the day side constantly searing hot and the night side constantly near absolute zero. But in 1965 it was discovered that the planet is only sort-of "locked," in a 3:2 orbital resonance that gives it three days every two years. Thus came the idea that a base would have to move, slowly, in order to stay out of the sun.

This is a subtrope of Base on Wheels.

Examples of Mercurial Base include:

Comic Books

The future arc of Elf Quest has Cauldron City on the Mercury-like planet Cauldron; it runs on rails that go around the polar region of the planet, and the main characters have to save it from sabotage.

Literature

Blue Mars by Kim Stanley Robinson has one of these on Mercury, of the "more plausible" variety. The city of Terminator (so called because it always rests on the planet's terminator) is mounted on a pair of planet-encircling tracks. The substantial thermal expansion due to direct solar exposure causes the tracks to continuously "push" the city away from the sunrise. The viewpoint character accidentally gets trapped outside.
- Robinson also wrote a short story, a murder mystery, set on a Mercury with a city on tracks.
As does Saturn's Children by Charles Stross; in this case, the protagonist's enemies tie her to the tracks and leave her for dead.
In The Thrawn Trilogy of the Star Wars Expanded Universe, there's the mining colony of Nomad City on the planet Nkllon. The city is made of huge platforms and grounded ships, sitting on top of 40 AT-AT walkers, and it uses gigantic umbrella-like ships which protected visiting ships on the way to the planet. Without the shade of an umbrella ship, any but the toughest spacecraft would soon be melted to slag. An Imperial attack damages the drives, threatening to leave the city stranded until dawn catches it; somewhat ironically, the threatened destruction of the base is a D-plot at best. Everyone aboard is easily evacuated after a minimum of drama, it's just Lando's latest ill-starred business venture that's in danger. It is not saved. It is quickly forgotten.
In the Alastair Reynolds Revelation Space book Absolution Gap, the theocratic society on the ice moon of Hela relies on travelling Cathedrals, which keep pace with the moon's rotation not for environmental but for religious reasons: to keep the vanishing gas giant within sight of their zealots.
The Periodic Table of Science Fiction, a collection of 118 short stories inspired by the elements, includes one of these in its mercury story.
The moving city in Alexander Tjurin's space opera "Final Station is Mercury".
There's an interesting inversion in the backstory of The Night Land. Before the Sun died completely, there was a point cities used to move west on rails to avoid the freezing night. It's unclear what happened when said cities reached the ocean.

Video Games

There's a mission in StarCraft II where the local sun turns into a nova. Playing the Terrans, the player has to move his base and his units to avoid the light of day which destroy units and buildings pretty quickly.
In Mass Effect 2 (Specifically, Lair of the Shadow Broker), the Shadow Broker's centre of operations is on a ship constantly traversing a planet at the terminator. The day side is baking hot, the night side freezes solid ten minutes after sunset, and the terminator itself is wracked by storms. This is actually a case of showing their work, because this is exactly what would happen along the boundary of such an extreme temperature differential.