Mines should always have a small stone or clay byproduct...their digging something.
They currently do. It seems to be about 1 stone per 3 coal (not sure about the copper version). This might b e a sign that the devs should make it easier to see bonus production for mines.
Mines already produce stone along coal/copper. The need for stone goes down quite a lot during the game as it can only be used to make bricks that you don't need too much of, or training gear that is only relevant if you find a service building for bloodthirst. On the other hand the need for fuel is constant, so swapping those around would be a very bad idea. If mines produced coal but had a chance to produce some copper as well without having copper nodes there would make gathering coal at least a little bit easier.
Mines are a bit underpowered and expensive at the moment, unless you have spare beavers to stock them with or lucky 3+ tile spawns.
Also, it's quite common to run out of sources of clay and stone in the later stages of the game if you're conservative with your exploration.
Finally, copper feels a bit underpowered right now, at least relative to sources of crystalized dew, so building a mine on top of 2 tiles of copper always feels awful. Especially if the ore is far away from your storage. Oftentimes it's not worth investing the 3 parts into building it and you'd rather trade for the materials you need.
By making mines be primarily stone and have coal or copper be byproducts of the mining operation (since they are deposits in rocky/earthy ground) then it would help solve both the copper/dew imbalance and the occasional stone droughts.
So I would one-up this suggestion by not having them be Coal mines with incidental stone, but Stone mines with incidental coal. The actual production rate is up for balance exploration, but it should improve the overall experience.
Maybe slightly more stone/worker than a gatherer's camp (25% more, or possibly 50% if it's not too strong) with a 25% chance to drop either coal or copper per deposit under the mine. So a 3 coal tile mine would drop coal 75% of the time, for example, while 2+1 would have 50% of one and 25% of the other.
This method would also solve the issue existing right now of having mixed deposits and incentivizing the player to withhold production of the resource with the fewer tiles. This is also an issue that an inexperienced or less skilled player is likely not aware of so it is generally further nerfing mines due to lack of knowledge and complicates the learning curve.