The mining rules could be adjusted for stone or other non-metal things you can mine, at whatever frequency you want them to occur. (I am assuming you’re talking about the mining rules I wrote in the downloads section; if not, the rest of this post will not make very much sense!)
The things you would want to adjust, to start with, would be the Mineral Veins Per Hex; if you want metal and stone to be rare, you don’t have to change this at all, but if you want metal to remain equally common you would need to increase the number of veins per hex. The amount that you would need to increase them depends on the next step; you’d want to make yourself a quick binary table, just two results on it. Whenever a vein is found, it’s either metal or stone. If metal, use the existing rules from there on; if stone, go further down this post.
To maintain the same level of metal, you’d need to divide the amount of veins found per hex by the chance of being metal. For example, if 50% of veins found are metal, you’d need to divide by 1/2 (that is, doubling the number of veins found) to make sure that the average amount of metal is still found. If, however, you just want metal and stone to be less common, maybe there’s just half as much metal!
If you have decided you have found a vein of stone, for the purpose of abstraction and not needing to write any more rules right now, I would then reflavor the existing rules to figure out how much stone you actually extract in a week of work. One mining team gets you a base of 40 gp of work per week. Roll for quality of the vein of stone as normal, and multiply the quality as a percentage by the base work to find out how much GP value of stone can be extracted per week. Stone doesn’t need to be smelted, so this is really the only thing you need to do for it.
If you want to, however, you could reflavor the common/precious metals table to be different kinds of stone, and use that as a base to get further customization in your veins of stone. If you do this, consider the amount of stone of ore mined per week as a percentage of the weight of the load of trade item as a modifier to the amount of stone mined. (That is, for common metals, 100 stone/week [tin] is baseline, 200 stone/week [iron] is double normal productivity; while for precious metals, 4 stone/week [electrum] is baseline, while 32 stone/week [mithril] is 8x normal productivity).
(PS - I wrote this reflavoring addition in the text box and not a spreadsheet like I would normally do, so there is a possibility of a math error somewhere!)