I've spent some time thinking about it, and here is a system I think would work well.
Basic Game
In the most basic game, there are three types of coins, the Copper Piece, the Silver Piece, which is worth one hundred copper, and the Gold Piece, which is worth 100 Silver. Each coin is roughly the size of a U.S. Quarter, and fifty coins weigh 1 pound.
Most campaign settings have some form of great fallen empire in their history, and it's simple to assume that they set down the weights and common values of currency, and those traditions have held in most of the world.
Advanced Options
Advanced coinage can be layered on to the level of complexity a group likes.
Bullion: Adds bars and bricks to the game. These are commonly used for investment, storage, and large purchases. A bar is worth 10 pieces, and weighs ten times as much. In fact, coins are called pieces because bars were segmented and could be broken apart. That practice has fallen into antiquity, but a Silver Piece is still worth 1 tenth of a Silver Bar. Bricks weight 20 pounds and are worth 1000 pieces.
Other Materials: Adds Copper-Nickel and Electrum Pieces to your game. These are the half-silver and half-gold respectively. These are rarely minted as anything but coins, and are used because they represent practical denominations.
Platinum: Platinum is extremely rare, and moreover, has a melting point higher than most mints are capable of reaching. Dwarven kingdoms are the only ones that commonly mint platinum coins. Some Dwarves pride themselves of absolutely identical coins. Others make each platinum coin an individual work of art. Additionally, some non-dwarven kingdoms have been known to put a powerful wizard in charge of the mint explicitly to produce this rare coin. A Platinum Coin is worth 50 gold pieces.
Named Coins: Different regions have different names for each type of coin. For example, copper, silver, and gold pieces could be Farthings, Marks, and Crowns respectively.
Ancient Coins: The fallen empire left a legacy in money laying in old vaults and dragon's hoards. These coins are actually rectangular segments broken off of bars. Players might find partial bars containing anywhere from two to nine intact segments.
Exchange Rates: Not all kingdoms mint consistently weighted coins, and some regions are notorious for having their coins clipped or shaven. Coinage from these places may only fetch 90% of their value. On the other hand, some kingdoms are known for having very precise mints, with coins having intricate ridges and artwork to prevent cliping and shaving. These coins can be worth up to 110% of their value.