If the rock is swinging, the wind is blowing.
If the rock casts a shadow, the sun is shining.
If the rock does not cast a shadow and is not wet, the sky is cloudy.
If the rock is difficult to see, it is foggy.
If the rock is white, it is snowing.
If the rock is coated with ice, there is a frost.
If the ice is thick, it's a heavy frost.
If the rock is bouncing, there is an earthquake.
If the rock is under water, there is a flood.
If the rock is warm, it is sunny.
If the rock is missing, there was a tornado (or the Rogue stole it).
If the rock is wet and swinging violently, there is a hurricane.
If the rock can be felt but not seen, it is night time.
If the rock has white splats on it, watch out for birds.
If there are two rocks, stop drinking, you are drunk.
If the rock is glowing, get to a fallout shelter.
Weather Stones have been "prognosticating" the current conditions for as long as…well, probably as long as there have been rocks. They grace the sites of Boy Scout camps, cabins, hunting lodges, military bases, parks, etc. Essentially, anywhere you’ll find people spending prolonged periods of time outside together, there's a chance you’ll find a Weather Stone nearby. Certainly there was always one, either mounted or hung from a tripod, at every Scout camp I attended as a kid. So, at the very least, I can confirm they've existed since the '90s. Some are homemade, some store-bought, but they all tend to be similar, regardless of their origins. Likewise, the instructions (incantation?) may vary from place to place, but that ancient humor still abides... no matter how old it actually is. Perhaps they're a bit kitschy, or at the very least, an old and (possibly) overplayed joke, but they might can actually bring something to the tabletop. The Weather Stone in your tabletop games? It can be done!
Obviously, bringing these rocks directly into tabletop games would be the easiest way to include them. They could be hanging outside the local inn (or spaceport) your players have chosen to visit, providing an easy (and literal) touchstone to the common world every one is trying to escape together. If your players have heard of them previously, they may even start consulting it regularly, should they continue visiting the locale... even if it doesn’t really tell them anything. You might even consider giving the stone actual weather-predicting or controlling powers, although you should be careful with how dangling powers like within PC's reach might affect the power balance of the game. In any case, this would be the simplest way to incorporate a Weather Stone.
Another way to bring the idea of Weather Stones to your tabletop games is as a source of inspiration. You can always provide something similar as a local or planetary custom that might seem unusual to the PCs. Perhaps everyone brings a twig to social gatherings for the “fire” - even if the hosts have grown past the need of having a fire, stove or fire pit, it could still be tradition to bring twigs to group gatherings. Maybe everyone in a region wears a particular kind of garment from when climate conditions were different or less-controlled. Or perhaps everybody just wants their own personal rock to wind a piece of string around... doesn’t everybody? This tradition could be a source of local pride, a current custom, or an old way that's slowly being rejected by a new generation. One of the great things about Weather Stones is that they're relatively easy to describe. There's a rock, something to hang it from (and with), a list of instructions, and not much else. Likewise, you should keep your own selections simple. Just including these rocks (or an equivalent tradition) in a session, even without adding much to them, creates a unique situation players will find memorable. If you can find one thing every week to spend a small bit of well-crafted time on, you’re sure to do well with making player memories.
Perhaps this post gives an awful lot of credit to something that's just a stone-cold simple gag. If you look just a stone's throw past the rock though, you can find an amazing amount of potential to add to your storytelling at the table or elsewhere. The important thing to remember is that your goal is to always bring something into games your players will appreciate, even when they're just one-off descriptions. It might seem strange at first, but details like this enhance your games in ways you can only appreciate with time and distance. Now if you’ll excuse me, my rock is showing a "20" - that must mean it's time for gaming. Until next time, enjoy your