Wooden cases are covered fully in the range of books onfurniture restoration available, but marble clocks are fairly unique to the restorer.
There is no magic chemical formula for restoring a black mirror finish to dulled and greying slate (actually a limestone). The correct method involves completely dismantling the case and re-polishing each individual piece with sophisticated polishing machinery. Dismantling can do more harm than good because the cases were often wired and cemented together very firmly, and re-polishing requires special skills. The greyness of the case is the result of years of oxidation, particularly as a result of the heat from Victorian fireplaces.
Some 'experts' suggest rubbing with fine abrasive paper but you should be very careful if you do this. Under the oxidised layer is often a coarse material which will not take a shine easily.
M&P's Slate Blacking is a compound made especially for this purpose and will normally produce an excellent finish unless the surface is very grey. To stain marble clocks, it is important to remember that only a small mount of blacking need be absorbed for it to work
For a sucessful application, apply the least amount possible. The product is so strong that the least amount absorbed by the surface should be sufficent to stain it. Having been left to soak in, remove all the surface with a cloth.
A final coat of Beeswax Polish should bring a deep black shine to the surface.
Many marble clocks had fine gilt lines engraved in the case. In most instances, this is in good condition but under years of grime. When the case has been blackened, waxed and polished, take a pointed instrument like a piece of sharpened pegwood and follow the lines. You should find the grime comes out leaving the original gold showing. If this fails to work, clean out the grooves thoroughly and gild with transfer gold leaf.