Do you have access to a smartphone (and permission to use it)? There are many implementations of dual n-back for the popular mobile operating systems. (I can't try any of those applications, so I cannot offer any advice on which ones are good/bad)
ETA: gwern's FAQ explores dual n-back well (and some lists some other mental exercises towards the bottom).
Do you specifically mean "nobody is at the register" or "greeting people at the register counts as brain-down-time?"
Because it's worth noting that whenever there is a person at the register, you have a valuable opportunity to improve your social skills.
Yes when nobody is at the register. I know its a great opportunity for improvement in social skills and in fact have noticed an improvement (though i have also noticed some things i find very annoying about some social conventions)
I was a cashier for a while with a shift that went from 5 PM to 11 PM, which took from "super busy" to almost dead. I used the late night time dead zones to work on writing projects. I don't know that I ever did anything all that productive with the few minutes of downtime you periodically got in busier sections.
I don't have recommendations for mental exercises. Are there mental exercises that you normally do, or are you assuming that mental exercises are valuable things you should be working on, and this seemed like a good place to do it? My belief is that 90% of the rationality that's worth learning is done by doing stuff, not thinking about stuff. (Or rather, thinking about stuff while you're doing it).
There's plenty I find annoying about social conventions, but I actually took the opportunity at the register to practice subverting them for comedic effect[1]. Learning when and how to skirt social rules is a valuable (and fun) skill. Although in truth, you should probably self-modify to treat social conventions like a game with interesting rules to learn - you're going to have to adapt to them one way or another. A lot of the time you can get away with subverting them if you project confidence and understand the meta-rules that led to them in the first place. But another lot of the time, you really, really need to be following the regular scripts or you're going to be shooting yourself in the foot and no amount of complaining about the irrationality of the world is going to help you.
[1] Comedy in particular is a valuable skill and the checkout line is a great place to practice it.
I'm in a similar boat; I work overnight at a well-known US gym for 8 hours, and the shift is so slow I'm allowed to pull out my laptop, books, phone, whatever I want. (No internet, though, except on my phone.)
That said, I recommend worldbuilding. I do it for my tabletop games, but you could just as easily do it in the modality of a novel. Of particular use would be the creation of the histories of countries and political systems; in this way, you could experiment with social conventions... perhaps some outside your own culture. You are, of course limited by your imagination... but only your imagination.
You could try doing the exercise I posted on here.. It's more fun when going around a circle, but it might be a nice way to pass the time and work on your idea-generating muscles.
As an add: I spend quite some time in the steam room, which has the added property of not being friendly to technical devices. What I do so far is to prep my memory system, and to think about single issues for a while. But I am happy to collect more suggestions, maybe a few from here are suitable for that too like mental arithmetics.
Why not practise mental arithmetic, like 45 times 23. It's not really rationality, but can't hurt. It's probably good for your brain somehow.
Or you could try doing fun pointless economics or physics calculations. If you're a cashier at a supermarket, you could calculate how far the chemical potential energy in a can of soup or whatever would propel it into the air, and do the calculation for as many products as you can find. Or figure out what proportion of the money that comes through you would have had to have stolen and invested twenty years ago on order to get your current salary. Or something like that. I dunno.
(Note: Here in Australia, cashier might have a different meaning. I hope I didn't offend you by implying you were a check out guy in a supermarket.)
So given my job as a cashier I have a lot of brain down time and was wondering if there are any mental exercises that one can do while alone to help improve my rationality.