Perhaps it was simply the quest for permanence/immortality that they used/admired stone. Stone was indeed the only thing which they could recognize of having weathered time, though I am not sure of what their understanding of time even was. Wood, leather, and any other material would eventually deteriorate, but stone would remain.

One could say "dirt" is permanent since there was always dirt right over there. Also, that yon grove of trees...always there. To them perhaps the state of dirt was fairly permanent, and unimportant that the individual dirt particles are not permanent. The same with the individual trees of yon grove. There are different trees, but the grove remains. Which level of permanence was important?

Since not many deities have been born of dirt, other than Mr. Clean's arch enemy Bob, I think stones were used/admired because they (as an individual entity) withstood time. Dirt, though always there, did not withstand time as an individual. Nor do trees.

Moral of the story:

When playing rock-paper-scissors,always use rock. Good ol' rock.

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Other than THAT, Mrs. Lincoln, how was the play?