The second part of an item's rarity, and in this case probably the most eye-catching one, is the decoration on the commemorative item.
Now it's probably best to start with the most basic - the flags and star theme. It's pretty clear that on their own, the flags and star are not rare at all - there are lots, and lots, and lots of cups with a pair of flags and a star (or sometimes just a pair of flags). However, not all flags are created equal. Consider, for example, this:
And contrast it with this:
It's pretty clear who's the better cup here - in fact, it's no contest. One has a very simple, almost casually painted pair of flags, while the other one has nice elaborate details and is done in a very visually pleasing manner. In this case I think the symmetry of the image helps. It makes the flags look well positioned and adds a certain level of gravitas to the cup commemorating an important victory.
Moving beyond the flags, there are a lot of different things that make it into decorations for cups. The most common aside from flags and stars are probably cherry blossoms (which represent the brief but visually impactful flower - like a soldier's life), chrysanthemums (representing the emperor), kiri (also a representation of the imperial throne), anchors for the navy, horses for cavalry and transport, various types of rifles, helmets, etc.
The positioning of various elements, and the composition of the wording, are pretty crucial in how an item looks. Some designs have obviously been honed over the years to create a visually pleasing arrangement. For example, the typical helmet + cherry blossoms + katana theme, like this:
The whole setting just works - it's obviously a stamped design, like a lot of designs on the smaller chokos tend to be (as opposed to the sakazuki designs which are more likely to be hand painted). There's a reason choko designs tend to be more elaborate and better balanced - leading to a more interesting looking cup.
The rarity factor really comes in two forms, and it's related to the final factor that I will talk about next - provenance. Basically, rarity in designs come from things they depict and the quality of that depiction. An obvious one is soldiers - especially soldiers of a specific type that have a unique design. For example, this guy:
A beautiful horse, with a well executed soldier. His face is clearly visible, with a serious expression. Nice details on the uniform. The whole image just works - and because of the level of detail and the skillfullness of the artist in creating it, it's really a one of a kind cup. I've handled some cups like these before, but this is far and away the most artistically pleasing looking soldier on a cup.
As I mentioned before, lacquered cups cost more back in the day, and they were also more likely to be well decorated. This set of cups that is beautifully done with an image of the bridge to the emperor's quarters in Tokyo is a nice example of elaborate urushi lacquer decoration:
The darker parts used to be silver colour, but because they used real silver in the paint it has darkened to a blackish colour. Can't polish it now to restore the luster, but one can imagine the cup must have been even more stunning when it was first made.
Not all beautiful cups with unique designs need to be elaborate. There's the balance issue I mentioned before that is often more important than mere elaboration of design. For example this very simple cup:
It's really very simple - a red lacquered cup with silver rays painted on it radiating from the center. Add some words, and that's it. But the effect is very stunning and looks great in person. I'm actually surprised this design isn't used more often - it couldn't be that difficult to make (compared with the castle cup above, for example) but it just looks like a million bucks.
The other kind of design rarity is really just in the objects themselves - not so much related to the beauty of the design, which I suppose are more subjective, but also whatever is being depicted. This is the factor that is somewhat related to the provenance issue - cups that have interesting provenance are more likely to have an interesting depiction, although not always true. For example, balloon unit cups:
What's there to say? It's a balloon unit cup. It's got a balloon on it. Balloon unit cups are very rare. Is it exceptionally pretty? Not particularly - you can easily see a field gun in the middle and then it becomes a fairly ordinary artillery unit cup. But no, it has a balloon on it, and balloon units are rare to say the least. You can say the same thing about tank units with tanks on the cups, or armored trains, or field tractors, or any kind of unit that are unusual and rare. They are more likely to have rare things on the cup itself depicted. Sometimes you can have a tank unit cup that only has a pair of flags on it. It happens.
Rare decorations though don't have to be because of the unit itself. For example, this signal unit cup:
That thing in the middle is a wire layer - it's a roll of wires that signal units used to lay down wires for communication. I stared at this cup for a long time trying to figure out what it was, because I've never seen it before. Then I remembered this scene from the movie "Enemy at the Gates" where one captured soldier was forced to push a thing that looks almost exactly like this to lay down a cut wire - and be shot by the sniper waiting in the process (to lure out the sniper). Well, that gave me the clue and lo and behold, I found pictures of Japanese units using something like this. So that was my answer. Signal unit cups are not rare, but a cup that has a wire layer on it? Yes, incredibly so.
Sometimes it's not even the item being depicted, but the style that matters, and this requires a bit of understanding of Japanese culture. For example, the recent ashtray that I put up:
The decoration might look somewhat crude - rudimentary, even. However, if you are interested in Japanese pottery in general, you'd know that one of the most famous makers of any kind of ceramics in Japan is the family of Raku Kichizaemon - it's one of those families that passes down the name from generation to generation. The person who made this was probably the 13th generation descendant of the family, who lived till 1944. The family is currently headed by the 15th generation. These guys are most famous for making tea bowls that were in the collection of daimyos and samurais, and the 13th generation was stuck in a period when interest in the tea ceremony was low due to a variety of factors. Making an ashtray with a military theme was obviously a departure for them - usually raku wares had no decoration, just a plain glaze. In fact I'd suspect that this is a very unique piece in the history of Japanese ceramics. I certainly have never come across another raku family imprinted ware that has a military theme on it, and because of the theme museums aren't likely to hold it. But that actually makes it rarer - there are lots of Raku Kichizaemon signed tea bowls, but how many bomber-decorated ashtrays with the Ministry of the Navy stamp on the back can you find?
The endless variation in these designs is partly what makes this a fun hobby for me - it's always interesting to encounter new cups even after ten years. I look forward to seeing new designs even now, which is saying something.