I know we can destroy them but I'd prefer another method.

What if everyday we had three groups requesting specific items (put it under quests) for a small amount of silver (less than 1 silver per item) but at least requesting 100 of the item to trade in?

For older players it encorages golem building to trade to other players or if they are getting over whelmed with a resource a two click way for getting rid of it (if it is one of the items to show up). For newer players it could mean a short influx of cash or choosing to trade away materials they don't need to players who might need the influx.

To help counter balance this sudden addition of silver I'd recomend in the same area offering three different types of matieral for at least 5 silver per item. Then there is still incentive to buy from other players but if you need an item you can't find there it might be avalible through the money sink.

The idea itself, I would see myself using, as a material-hungry newbie.
The numbers would need major tweaking though.

For what you get for selling - I'd perhaps not even award currency at all, but a system of virtual goods, such as tokens or tickets, that can be spent on fluff or moderately useful items, the reward should be like a consolation prize for otherwise completely useless stuff, not a direct benefit to make the rich richer.

As for buying - many materials are near impossible to find even at seveal 100S a piece. The price should not only not undercut players - it should be significantly higher. This system should be an absolute last resort, and encourage players to trade amongst themselves, rather than just buying everything from the game.
Much like the blueprints in the state store - there is the safety net, to fall back on just in case, but it'll be wildldy more advantageous for everybody involved to buy it off another player.

How much tweaking of the numbers? Perhaps 10 silver per item? 20? or perhaps have a teired system (a pound of food is five silver, easy building materials 10 silver, harder to find are 25 etc).

My concern with tokens is needing to find something to spend it on. (perhaps invites?)

For the selling part:

The reason for tokens is because the benefit is supposed to be minimal.
If there is any demand whatsoever for your item, selling it to other players should be more beneficial than simply deleting it.
At the same time, introducing easy quantities of additional silver is the exact opposite of a money sink, and also uncalled for.

For the buying part:

Quite the opposite of above. If there is any supply whatsoever, buying it from other players should be more beneficial than simply buying everything from the game.
If buying from the game is cheaper than from other players, and easier than gathering it yourself, then there's no reason to play the game anymore. Just throw money at it and be done.

So while both would be nice safety net options, they should both be discouraged and presented as a last resort, in my opinion.

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Another way to perhaps go about it, would be to simply make it a two-way system.

Say you just capped on stone, and have absolutely no use for it, so you donate 500 stone to the system, and at a 250:1 ratio, this award you 2 points.
The points in turn can be exchanged for 2 units of any desired material.

That sounds like it's very, very cheap - say, 2 quartz for 500 stone, but consider:
The least desireable of materials frequently sell for 1S a piece or less, so 250S for a piece of granite or quartz, would actually still be a very good price to buy at.

Now, the effect would be that even the least wanted materials get some value, if only for their ability to be turned into other materials, while making the more desirable materials a bit more accessible, and thus balancing the scales a bit, without clumping everything together.

It would also allow removal of excess materials, a controlled source of rarer materials, and while not being a moneysink, it won't bring more silve rinto the game either.

I don't know if we should be assigning exact values to items- That could really skew the market which is supposed to rise and fall with demand and supply. 

I propose a third idea which many websites use- a junk yard! 

Here we can exchange goods for items exclusive to the junk yard. Perhaps what the junkyard accepts could be a randomly chosen material, or could vary by month or week. Or even be on a point system like Saccharissa suggested (I like the skewed rates).

It would be something like "Do you want to exchange 150 stone for this SUPERAWESOMESHINYITEM?!"

The amount of materials to exchange should be kept somewhat large so it can truely be a place to exchange unwanted materials for coveted items; thus keeping the items in the junkyard somewhat rare enough that people would rather exchange for them than purchase them. This could also drive people to explore more. 

I like upping the amount needed to be given, and allowing to trade a huge amount for the item needed (especially at a very large amount needed to get points.) 

The problem I have with the junkyard is you still need to come up with items for it, whether they already exist or are totally brand new in order to make it any use.

Edit: just thought of something that would make this system still a safety net but still prioritize player tradeing. What if the area only accepts and sells certain items at a time that changes every 4 hours? Rare items need higher amounts of points or a higher trade rate, but the rate is not always the same. Have a sliding scale, so even if players try to price based on how much the game says its worth it could change right at the 4 hour mark (and not everyone would be able to trade the same stuff)

I was just thinking about a game I used to play, and a feature it had... gamesale value.
Basically, every item has a low gamesale value - simply selling the item back to the game deletes the item and gives you fast, if small amounts, of money.
More sought after items obviously were worth much more than this if sold to other players, but for junk items it was a convenient and easy way out.