Stable money does not make much sense.
First, what does stable money mean. Most people would agree what they mean is that if $5 today gets you a loaf of bread, that $5 will buy you a loaf of bread a year or so from now as well.
Let’s say you have a closed economic system which only produces bread, has 10 producers, and $100 in total. Those 10 producers create 20 loaves of bread, so a loaf of bread is $5. But what happens if they are hit by a meteor and 5 of the producers die. Now you still have $100 but only 10 loaves can be produced. So what would it mean to keep the price of money stable? If you still force a loaf of bread to be $5, after the first 10 loaves are bought, you’re out of loaves ans the remaining $50 have effectively inflated away to be worthless. Instability of currencies, therefore, is a natural consequence of imbalances between the supply of money and the production of goods and services.
The US has actually done a tremendous job keeping the USD fairly stable, by printing more money to account for growing economic output (due to a variety of reasons, technology being one of them).
That being said, there is always a bias towards low inflation as opposed to low deflation, because deflation can be an economy killer. If the value of cash increases with time (in other words, if $5 today buys you 1 loaf of bread, it will get you 2 loaves of bread a year from now), then the risk adjusted returns from any investment needs to be that much greater for an investor to invest.
Even a small percent or 2 of deflation can have an oversized effect that would lead to a significantly smaller economy.