Once upon a time there was a coin called "the mil". This was a "milli-dollar" coin, one thousandth of a dollar (one tenth of a cent). Inflation made it obsolete, and about the only vestige of it remaining in US society is in the signs on gas stations, where the price for gas almost always is listed
as something-dollars, something-cents, and 9/10 of a cent, per gallon.

In 1964 the US Mint stopped making quarters and dimes out of silver. Inflation had made silver too expensive for it to be used that way. Nowadays they are "sandwich" coins having copper cores, with a silvery cladding (maybe nickel?).

In 1982 the US Mint stopped making pennies out of mostly copper. Inflation had finally made copper too expensive for it to be used that way. Nowadays pennies are zinc, with a copper cladding.

Recently inflation has made even these pennies too expensive to keep making. Either a cheaper metal will have to be used, or they will have to be abandoned altogether, like the mil. I understand that nickels are starting to be too expensive to keep making, also. And considering how much copper still goes into making dimes and quarters, it seems to me that a significant coinage shift is about to become Needed.

SO:

The Quarter-Dollar will likely have to get a new composition. Possibly it will get its copper core replaced with zinc (that seems to me to be the simplest thing to do, using and combining existing Minting technologies).

Next, since a quarter-dollar is still too large a unit for various small items that you might buy, we still need something smaller than a quarter, but larger than the likely-soon-to-be-abandoned penny.

Note that if we are likely to abandon the nickel soon, also, then we have a dilemma in that you can't "make change" of a quarter into dimes only. This is why I'm suggesting we eliminate all three coins, pennies, nickels, and dimes, and replace them with just one new coin, as follows:

One sixteenth of a dollar is 6.25 cents. If there was a coin having this value, then exactly four of them would equal a quarter-dollar, just like four quarter-dollars equals a whole dollar. Making change from a quarter would be easy.

Since 6.25 cents is so close to the 5 cents of a nickel, we can immediately see that this coin will have to be made of some inexpensive metal, perhaps entirely from zinc, or maybe even aluminum. (Aluminum has been used for coinage for decades in various other countries.) Maybe even galvanized (zinc coated) iron could be used!

Two problems remain. First, if the coin has a value of 6.25 cents, then when we write down some amount of dollars and cents, we no longer can do this with precision using only two decimal places: One Dollar and One One-Sixteenth Dollar is $1.0625 --very cumbersome! We would need a simpler way to write it. I will suggest that we introduce the semi-colon (;) symbol into the numerical description, like so:

First, using quarters only:
$1.25 = $1;10
$1.50 = $1;20
$1.75 = $1;30
Note that when you have 4 quarters you don't need to write $1;40 because you can instead write $2.00

We extend the above for the Sixteenth-dollar coin:
$1.0625 = $1;01
$1.1250 = $1;02
$1.1875 = $1;03
$1.2500 = $1;04 = $1;10

Basically, and for small amounts of change, on the right of the semicolon we simply write down the QUANTITY of quarters and sixteenths. So if I have 1 dollar and 2 quarters and 3 sixteenths, I simply write $1;23 --it's a little more difficult if I have 10 sixteenths, and want to write it down; I have to mentally compute that 8 of those coins equals 2 quarters, so I would end up writing $0;22 because there are 2 sixteenths left over, after doing the conversion.

The hardest problem of all I have saved for last. "One Sixteenth of a Dollar" is a pretty long NAME. Even "sixteenth" is harder to pronounce than words like "quarter" or "nickel" or "penny". We need a simpler name for this proposed coin.

Old coins had the ability to be broken and the values were a fraction of the coin. this is where the term "pieces of eight" (1/8 of a doubloon) and "bits" ("shave/haircut, 2 bits" "2 bits, 4 bits, 6 bits, a dollar")
I rather stay with the base 10/5 system we have (maybe adding a 15 or 20 cent coin?) Some countries have changed their currency to this system. Baked & boned.

This is a bizarro, complicated workaround, when it would be far simpler to just drop the penny from circulation. 1/16 of a dollar is no more convenient than 1/20. Then again, I guess this is the home of bizarro, complicated workarounds.

They could make all coins semicircular, dividing the production costs in half but retaining the original value. Keeps coins from rolling on the ground when dropped, plus you'd have the fun of trying to find two half-nickels that fit together perfectly.

While this would probably be a great mental exercise for learning and using fractions, in a society which uses a decimal or base-10 system it seems rather counter-intuitive. While you have ably demonstrated the benefit of "sixteenth" notation to commerce, I see no practical value for its use in mathematics and science, which would likely doom it to failure. However, I do admire your highly optimistic point of view that a nation which refuses to embrace the globally-accepted metric system would flock to a monetary system based on a small disc-shaped metal amalgam worth 25 cents.

As for difficulty in finding a name for it, that's the easiest part. The latin form of sixteenth would be sextusdecimus, or thereabouts, so you could call the coin a "sextus" for short.

Why not move the monetary system to
base 12 or 16 in the process of making
the
conversion? My vote is for 12. That
way it
is easy to do both 1/2 bux, 1/4 bux
and
1/3 bux while avoiding the messy
notation. People will just have to learn
that $1.30 is actually a dollar and a
quarter whereas $1.60 is a dollar and a
half. And a dollar and 1/3 is $1.40.

To make the transition easier, lets just
abandon the decimal system alltogeher.
We should be able to move behind that
silly vestige of having 10 fingers.
Perhaps to make counting easier for
those who need to rely on manual
counting, there could be a market for
artificial sixth digits to be added to
one's hands.

You could revert to English 'old' money - A pound being equal to 20 shillings, and each shilling equal to 12 pence. Then there's the half crown, equal to two shillings and sixpence, and guinea (being 21 shillings). Further, hapney, thruppny and sixpence pieces could be added to the mix in order to add to the general confusion.

[Goesta Berling], Base 12 has its niceness, but Base 16 is more relevant to the Computer Age --and translating between Base 4 and Base 16 is very easy.

So, I personally would prefer to see a shift to Base 4 for money. Then not only would there be quarter-dollars and sixteenth-dollars, but there would be 4-dollar bills, 16-dollar bills, 64-dollar bills, and so on. The Mint would actually end up with fewer total denominations, and one thing it has wanted to do for decades, reduce the number of 1-dollar-bills in circulation, would be easier as 4-dollar-bills became common.

I should mention that if we stop making pennies, nickels, and dimes, and replace them with "steens", then cash register drawers suddenly have some spare coin slots. One other thing the Mint has been wanting to do is get dollar coins into wider circulation (it's another way to reduce printing dollar-bills; coins last lots longer), but this has been stymied by cash register drawers not having enough spaces. This Idea would fix that problem.