This is interesting and you cannot tell what will happen. All you can do is watch and see and hope. In the meanwhile, I don't think it will do any harm to save your pennies in a place where they won't be a bother to anyone. Some things that you never thought would happen are happening now and things once thought of no/little value are now on demand and are valuable. It's good if you can see opportunity and seize it.
Seriously, have you looked at the performance of copper mining stocks and the price of copper in the last few years? The copper collapse is worse than gold. The current Chinese economic slowdown or stall (pick one) exasperates the situation. I bailed on $FCX years ago and see absolutely no reason to get back in. There are too many other good opportunities to pick from.
When it comes to the penny, it may be wise to stash away pre-1982 pennies. These are the ones that are 95% copper. Obviously, the penny will always hold a face value of 1 cent. When it comes to the topic of stashing 95% copper pennies away, we are talking about the melt value, or the copper, of the penny. If copper prices rise, you would theoretically be able to sell off your pre-1982 pennies for over a cent each, because the industry would be more interested in the copper of the penny then the actual face value. The same goes for the nickel. The nickel holds a 5 cent face value, but the nickel is also 75% copper. If copper skyrockets one day, the metal content inside of the nickel will be worth more than the actual face value. So, to answer the question, it is possible for the copper content in a penny to be worth $1.00 someday, if the penny is pre-1982.
You can keep track of the melt value of your coins daily on Coinflation.com It's a good list to keep handy when you sort through your change dish/jar/bucket before you head out to pay your tax bill.
It sounds like a great selling point for the investment approach, but I'm not sure whether I'd start stashing lots of coins in the event it might happen. It's not the first time this concept has existed. I recall in the past some British coins were made with a bit of silver in the past, and over time the silver content decreased, even though the denomination of the coin stayed the same. So if you were smart enough to collect some of these coins you'd make a bit of money potentially. But you would probably have to wait a while for it to play out.