Some background information.
First of all one should notice, that Golgi staining belongs to so-called morphological types of stainings in neuroscience, where the actual anatomy of different neurons is revealed (compared to other techniques, like Ca-imaging or potential imaging).
Second, Golgi staining is a type of silver staining, there the sedimentation of silver or its salts (here: silver chromate) reveals the morphological traits of the cels.
And, third, Golgi staining is applied to fixed preparation. This means that the tissue (normally a brain slice) is pre-treated (here: with formol) to kill all cells and arrest every biological process.
What is known about the targets.
The common description of the target is quoted as "a limited number of cells at random in their entirety". This is the essence of Golgi staining:
Only single cells are stained, therefore there is no impediment from adjacent cell staining while deriving the morphological structure of the cells (very important for light microscopy where you have integrated input from different depths).
The cells are stained randomly, there is no known preference among neuronal cells for Golgi staining and I haven't seen any other types of cells in CNS stained with Golgi, therefore it is quite specific for neuronal cells (and leaving macro- and microglia, astrocytes etc. intact).
The cells are stained in their entirety, meaning that the complete cell is stained very nicely, showing detailed arborisation of dendritic tree, that was very important in studying of Purkinje cells in cerebellum.
Are larger neurons more likely to be stained? Are specific cell types more susceptible than others?
Golgi staining is used mostly for brain slices (I have never seen or heard its application for other tissues). Traditionally one of the biggest cells here are pyramid neurons (NA, ACh-ergic) and one of the smallest are interneurons (often GABA-ergic) -- both are amenable to Golgi staining (reference) and there is no seemingly clusterization of stained cells by their size or transmitter type.
What is preventing us from using the advanced molecular biology techniques to understand the process?
I can name several reasons for this:
Since Golgi staining is applied to fixed preparation the tissue is already "damaged" (formol leads to dessication of cells and shrumping), therefore it is difficult to use some fine mollecular biology methods to investigate these tissues.
There is no way to tell which cells get stained beforehand. And as long as the microcrystallisation of silver chromate is started it can't be (easily) stopped and reversed. Therefore it is difficult to look at what caused the staining afterwards, then the whole cell is impregnated with silver.
I think there were no real attempts to crack the mistery of this staining: how intersting it might be, this seems to be an interdisciplinary question on the brink between biology and chemistry. So, maybe one day somebody will look into it and explain everything.