[Note: This is a revised and amended version of an original post which, as several critics pointed out, failed to address the legal argument for using platinum.]
The only sort of coin that can address the debt problem is, of course, a fiduciary or token coin, worth more than its cost of production; moreover, any unnecessary expense incurred in making such a coin, by making it out of a precious metal or otherwise, merely serves to reduce the difference between its nominal value and its cost, detracting correspondingly from the profit or seigniorage the Treasury might gain by depositing it at the Fed.
True, if the coin were intended to circulate along with many others of the same sort, its cost of production would serve as a deterrent to counterfeiters, though the deterrent would still be small in proportion to the coin's fiduciary component. No one, however, imagines that a trillion-dollar coin will actually circulate; and even if it did it would do so only after the most exacting of inspections determined that it was the real McCoy rather than some aftermarket fake. In short, for the Treasury to bother making it out of anything so precious–and so intractable–as Platinum (the high melting point and hardness of which make it a very difficult material for fashioning high-quality and therefore counterfeit-resistant engravings) would be perfectly foolish. Given the coin's purpose, it would be far wiser to fashion it out of the same junk now used to make pennies, or for that matter out of plastic, or out of cardboard made from recycled copies of the Congressional Record.
Indeed, if the thing is never to leave the Federal Reserve's vault, it might as well consist of nothing more than a cover from one of those little ice-cream tubs–you know, the ones with the wooden spoon underneath–on which Congressman
Walden Nadler has scribbled the words "$1 Trillion" along with some appropriate legend. In case the good Congressman is reading this, perhaps he will consider my proposal for such. It is: "In Idiots We Trust."
Addendum: On Facebook Jacob Levy observes that the proposal to use platinum is solely due to the fact that enabling legislation already exists for making a trillion-dollar coin using that metal. That is of course true; but the significance of this "loophole" seems to me much exaggerated, for it only allows the Treasury to "mint and issue platinum bullion coins and proof platinum coins." The rub is this: a "bullion coin"" is, according to the U.S. Mint's own glossary, a "precious metal coin traded at current bullion prices," which is to say, traded at its bullion value rather than its face value, whatever the latter may be. So the Treasury could not gain much if any seigniorage by exploiting the "bullion coin" component of 31 USC 5112. A "proof platinum coin" might, on the other hand, be made of any sort of platinum alloy, which needn't correspond to that of circulating coins of which the proofs are special representatives. So far, so good. The rub here is that a proof coin itself is not a circulating coin, and hence is not legal tender, so that the Fed would again have no business crediting it at its (arbitrary) face value assuming that it finds a way to give any credit for it at all.
On the other hand, the Fed might conceivably purchase, and the Mint might sell it, a cardboard commemorative or proof piece just as readily as it might a platinum (or platinum-alloy) one, and do so for an arbitrarily high price–just as the Fed has purchased all kinds of securities for more than their presumed market value. The specific denomination of the piece needn't matter; indeed it needn't even bear such. And neither need the Fed deal directly with the Treasury: the Treasury might, for instance, sell the commemorative to a private person–say, a large financial firm or insurance company or auto manufacturer–which could in turn seek Fed support on the grounds that it is Too Big To Fail and that the purchase, although it seemed like a good idea at the time, will otherwise force it into bankruptcy.
So far as I know, there is no rule concerning what prices the Mint may accept in outright sales of its commemorative and proof pieces, whatever those may be made from. As for the rest, the precedents are perfectly well established. (Smiley face added to assist the humor-impaired.)