As a DM, I always felt bad for the poor copper piece. Always the one to be tossed aside for something better. Always scoffed at when offered as payment. Most likely to be thrown down a hallway to check for trapped floors.
As adventurers become more and more powerful their standards of treasure become higher and only a certain caliber of monetary gain is suitable. Only gold and above will suffice for the well-dressed, high level adventurer. So where does that leave the poor copper piece? The bottom of a beggar's cup or stuck to the sole of an adventurer's boot that's where.
I believe the disdain that most players have for the copper piece, and to a lesser extent the silver piece, is the fact that gold is the standard unit of money in most games. Meaning that everything is priced based on a gold piece value and therefore, why carry copper pieces as they are next to worthless in the greater scheme of things.
What I have done in the past is taken a page from the sage, Gary Gygax to you and me, and adopted a different money system based loosely on that presented in Greyhawk Adventures - Saga of Old City. In which, he expanded the money system to include several other types of metal coins. I have taken that basic premise, changed some names to protect the innocent, simplified the exchange rate and changed the basic unit of coinage from the Gold Piece to the Bronze Piece.
The new money system looks like this:
Iron Blight (ip) Lowest value coin
Brass Cutter (bp) 10 Iron Pieces
Bronze Filch (bz) 10 Brass Pieces
Copper Swift (cp) 10 Bronze Pieces
Silver Talon (sp) 10 Copper Pieces
Electrum Galleon (ep) 10 Silver Pieces
Gold Spire (gp) 10 Electrum Pieces
Platinum Raven (pp) 10 Gold Pieces
Mithril Corsair (mp) 10 Platinum Pieces
Adamant Drake (ap) 10 Mithril Pieces
So as you can see, now when they find a copper piece, they won't be so quick to toss it aside. Sure, it might just replace the lowly valued Copper and Silver pieces with differently named Iron and Brass pieces, but at least Iron and Brass don't seem as valuable as Copper and Silver. If you left a round piece of iron laying on the ground, you wouldn't think twice about it.