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I'm writing a story that takes place entirely underwater, mostly in a deep sea town and shallower city. Both are underwater and populated by beings who can survive underwater.

My thinking is that they send messages to each other by taking a soft clay-like material (let's call it substance x until someone thinks of what it should be) and writing on it, then giving it to a messenger. The messenger takes this wad of substance x with the writing on it and delivers it to the recipient.

When they are done the wad is given back, the letters are smushed away, and it is reused. What material should this be?

It needs to be something that can be made of substances found in the ocean, can be made underwater, doesn't dissolve into the saltwater, lasts a fairly long time, and is inexpensive.

Is there a real-world substance that meets all these requirements, or am I gonna have to make something up?

What you described is an old invention of Wax tablet.

Spermaceti from sperm whales is just one of many waxes your underwater people could use. Some of the fishes also contains significant amount of waxes. And if you need only short-term solution, animal fat would also do the job.

I couldn't find any seaweed that secrete waxes, so I hope your underwater people are not vegan.

You don't have to go back in time very far to reach a point where marine-derived hydrocarbons pretty much made the world go round. The catch being that you have to kill rather intelligent, large and dangerous predators who share your environment.

Enter the sperm whale, who's head is filled with an extremely useful collection of waxes and oils which maybe extracted and refined to produce exactly the sort of material you'd be interested in. By way of a bonus, it also produces another kind of oily wax, ambergris. This is much rarer, so you would only use it for special occasions, but it would still suffice.

In both cases, you want to keep your message tablets safe in a heavy container or framework, as the waxes would be lighter than water and quickly lost if dropped.

Waxy molecules are found in various marine plants and animals in smaller quantities, so you don't have to wave your hands too hard to invent something slightly more eaily and ethically sourceable. The main waxy component of spermaceti, cetyl palmitate is found in some corals. I'm sure you could conhjure up a wax coral or sponge that produced it in usefully harvestable quantities

Braids of seaweed.

The ancient Inca used knotted fibers, or quipu, to record events and information. With enough effort, seaweed could be manipulated (slicing then braiding) into thin cords, which could then function in a similar way as quipu. I wouldn't be sure about the lifespan of these cords -- it depends on the manufacturing process -- but these makeshift quipu could have knots tied or untied, and would be relatively light or even buoyant in water.

(Not sure if this idea is dumb...)

Sandstone.

Sandstone is a sedimentary rock and is common. It is soft and easily carved with just about any tool or even a stick. Depicted is graffiti on the sandstone of one of the arches of arches national park. A google turns up many images of graffiti on sandstone (all copyright images); some of these graffitos are ancient.

Sandstone would be available under the surface as would stones or shells adequate to mark it. A sandstone tablet would be a fine medium to leave messages.