I've done a fair deal of spelunking (caving) and one of the first things you realize when you've visited several caves is that their inner air temperature tends to be at roughly the yearly average of the outside temperatures. They might produce their own climatic system if they're deep enough or have any connection with the outside temperatures, like both end openings or a connected body of water, but the deeper you go, the more isolated from the more rapid outside changes you are and the temperatures stabilize. This could still mean different areas of the caves might have different temperatures, even have the heater or the freezer effect in some places, but deep enough, they tend to be stable all year long.
So if the caves here on the Earth have anything to teach us is that the thermal dynamics of a well isolated system do indeed work the way the thermodynamic theories teach us; that any body of mass would try to be in a constant thermodynamic equilibrium with its surroundings, and the more mass you throw at it (or surround it with) the longer the temperatures will be, for all intents and purposes, considered stable. This should go the same for any celestial body, Mercury not being any exception.
Now, the celestial body might have a lot longer (or shorter) days, years, e.t.c. and a lot more dynamic external environment than the relatively mild changes here on Earth (comparatively speaking), so the depths at which you would find reasonably stable environment might vary substantially, even cave to cave depending on their topography and connection with outside environment, but given enough caves to choose from and perhaps some reengineering of their less stable climatic influencers, this should be perfectly doable. If that would be considered convenient enough, that's another matter tho.
There is however one big reservation in all of this. If the celestial body or the area where you wanted to occupy a cave is still geologically active, the caves might be even more active than their surroundings. Caves can form by tectonic shifts, geothermal activity, erosion, impacts or even just by cooling down and solidifying. Most of these processes last a long time and are difficult to predict, let alone control in any way. And trust me, caves are not a friendly environment to be in even with mild disturbance. I've been roughly one kilometer deep in one cave (we have thousands in my country) once, when a really small tremor happened that likely wasn't even noticed by most people outside. And even though I'm not claustrophobic, I did experience first hand then what it must mean to be.
Walls of caves here on Earth (at least the interesting to visit ones) are mostly made of limestone sedimentation, which is rather brittle even if it doesn't look like that with millions of years of being polished by wind and water, or cave topography slowly grows by sedimentation. And the walls of volcanic chimneys are often glass of once molten silicates. The problem is also what you usually don't see, which is the porosity of the whole cave system you're in. The larger passages are obvious, but most caves that were first formed by tectonic shifts of upper crust layers might literally have millions of smaller cracks around them and even enormous rock boulders suspended above the larger passages by brittle, unstable rock that might give in at any time and close access. So the caves anywhere else might be just as fragile and the processes that made them still active. And isolation comes with its own disadvantages as well. No radio communication will reach the surface unless you laid a cable with a transmitter on the surface.
Anyway, I thought to describe a bit what else might be a problem, because the stability of inner climate really isn't the most problematic. It could possibly be even artificially controlled a whole lot easier than engineering around the problem of the stability of the cave system you'd build your base (spaceport?) in.
