The first element of an item's rarity has to be the material with which it's made. Quite simply, it's the stuff that the cup is made of, and roughly speaking, from rarest to least rare, I'd put them in the following order:
Aluminium, Bakelite, Silver, Silver alloy, Tin, Pewter, Metal, Wood, Porcelain
It's easy to forget now with aluminium (aluminum) being so common, but back in the day it was an expensive metal. It eventually became cheaper when Charles M. Hall discovered a new process that made it a lot easier to produce aluminium, but that took some time and it seems like Japan still retained some value for aluminium. I've only ever found a couple cups made with aluminium, and they're interesting, to say the least.
The cups are really cute - they ring if you tap them, and they have distinctive sound. They're light (aluminium is light after all) and very interesting to handle. The metal is also fairly tough, so they survive. They were just made in such small numbers that they are rare to see.
Then the next up are bakelite cups - they are about as rare as aluminium, but they are usually less interesting in design, probably because making stuff with bakelite was not particularly easy to work with compared with metal. Usually they are small choko - 2" cups with simple, stamped designs.
Then we have the traditional, precious metal cups made with silver. Until pretty recently silver cups were given out by the Japanese government for people who lived to 100 years - but they changed that law because there were too many of them and silver was getting too expensive. In any case, we still see a lot of silver cups out there in random antique markets for various things - birthdays, weddings, etc.
Generally there were two types of silver cups - government issued ones and private ones. Government issued ones were for official purposes - military exercises, medal awards, official events, that sort of thing. This cup I'm selling currently is for someone participating in the enthronement ceremony of the Showa emperor, and given by the Minister of the Army. There were also these Golden Kite sets that were given out, I believe in lieu of pension payments as pensions were canceled after the war was over and the Golden Kite was abolished.
Cups for private individuals can come in all kinds of shapes and sizes. For example, I have this one set that I sold some years ago with a beautiful torpedo design for a person who worked in the Sasebo Arsenal.
Or this incredibly rare cup given by Lieutenant General Yokoyama in 1943 to a subordinate for participation in the fighting around Wuhan at the time. This cup was made in the city of Hankou in China and these late-war items are incredibly rare.
Silver of course is rare just because - it's worth something on its own. Even now, silver cups with no inscription can still sell for decent amounts because the silver alone is worth something. Since Japanese silverware, especially if they're stamped "pure silver" is very high purity, they never come cheap. Coupled with the fact that the cups are often decorated well because silver cups are expensive to make, so people were willing to invest a bit more in making it pretty, so you have often very nice collector's items on your hand when you see a silver cup.
The next in line for rarity are pewter-ware. Pewter, or tin (sometimes you get almost pure time items) was traditionally used for tableware. Nobody minded the lead back then, because nobody knew it would poison you. It was cheaper than silver and since these days you are not likely to be buying these items to actually drink out of, the lead isn't a problem. Pewter is easier to work with than silver, and also can come with nice designs. For example this cup I'm selling right now was given to someone on the battleship Nagato for winning the annual gunnery competition.
Both silver and pewter cups tend to be more for special occasions - while there are some for the usual "discharge" or "leaving the service", more often they are for things like these - special award or recognition of some kind, or special events. It makes sense. These were not cheap materials to use.
The next in line are wood cups - lacquered cups, for the most part, although once in a long while there are bare wooden ones without the lacquer paint. Lacquered cups, like silver and pewter cups, can be for official functions or private ones, although of course they are less rare and so the events tend to be more mundane. At the same time though, because it was relatively easy to decorated a lacquer cup compared with working metals, oftentimes the best, most richly decorated cups I find are lacquered cups. They really run the gamut - a nice picture of an eagle, depiction of people, world map like the cruise map cup I sold recently, etc. The best made lacquered wood cups are works of art, really. They were also not that easy to make - the wood was carved from bigger pieces, so it took considerable amount of time and effort to make. It's not as if the cups were made by machine just boring down the wood either - traditional lacquered items were made by hand. It's no wonder that they were expensive compared to porcelain cups. When done right, the results are stunning, like this rising sun cup that I really like.
Finally, in terms of material rarity, the most common are porcelain items. You can probably guess that, because most of the items that survive are porcelain. Porcelain has advantages - they're easy to clean, they are easy to decorate, so interesting pictures can be painted on them, and they were easy to make. Using a wheel to throw a cup is a lot faster than making a lacquered wood cup, for example - a skilled artisan can do a porcelain sakazuki in under a minute, then decorate and fire it. A wooden cup cannot be completed nearly that quickly. At the same time, because porcelain breaks easily, a lot of cups we get now have small or large damages on them. Sometimes they're very minor - a little chip or a hairline crack. Other times they're serious. Nevertheless, the variety of designs and the intricacy of details on porcelain cups can be unrivaled.
The main differentiator between various types of porcelain ware is down to the quality of the worksmanship. The roughest items can be pretty rough looking - a quickly painted flag with a few words. The more detailed items can be just as beautiful as any other cup - like the soldier cup I use for my homepage, or sometimes various hand painted items. There are also different tricks makers use - photo transfer, over-and-under glaze, etc, that make the cups look more interesting. Thematic variation is something I'll talk about next in terms of rarity, so I'll leave you with this cup of koi fishes overlaid with gold decoration for a victory commemoration.