In the Timaeus Plato assigns the cube to earth. The reason he gives is steadiness: earth is the least mobile of the elements, and of the five solids the cube has the most secure bases, sitting flat on whichever face it is set down. As with the other assignments this is an argument from fitness rather than from evidence, and it is a fact about what Plato wrote rather than a fact about soil.
The cube occupies a special position in his scheme. Plato builds the faces of the solids out of two elementary right triangles. The tetrahedron, octahedron and icosahedron all have triangular faces made from the half-equilateral triangle, and because they share a building block they can be dismantled and recombined into one another: fire, air and water pass back and forth. The cube alone is built from the half-square, and Plato states plainly that earth therefore cannot be turned into any other element. Break it up as you like and it reassembles as earth.
That is an unusually specific piece of theorising, and it is the nearest thing in the passage to a testable proposal. It is also wrong, which is worth saying without embarrassment. Matter is not composed of nested right triangles, and the four elements of the ancient scheme are not elements in the modern sense at all. What makes the Timaeus interesting here is not that it reached the right answer but that it attempted a geometrical account of substance, roughly two thousand years before there was any means of testing one.
Two attributions are commonly overstated. The four-element scheme is not Plato's invention; it is usually traced to Empedocles in the fifth century BC, and Plato inherited it. The five solids are not Plato's discovery either. What the dialogue contributed was the marriage of the two: elements given shapes, and shapes given a mechanism.