A brief history of currency
The Ancient Way: Money has the color of gold
For a long time, 'money' was only something that could be exchanged for value. This is more or less and extension of the barter system with something that is widely recognized as having a value among many cultures and regions. Gold was the one natural commodity that fulfilled this best on Earth.
Gold has several properties that make it useful as a currency. First, and perhaps most importantly, is that it is visually pleasing and distinctive. Nothing shines quite like gold. Second, it is durable; that is, it is not used up. Spices were consumed, and more practical metals like iron were beaten into swords and plowshares. Gold was never 'used up' on anything practical; it was kept and displayed as a decoration (and expression of wealth). Third, it was imperishable. Gold famously does not rust, barely tarnishes, and generally needs only a rub with water and/or oil to polish back to a fine shine. Fourth, everyone knew what it was. Even if you have something that aught to be valuable, if you are trading it in a distant place the potential customers for this product must also recognize its value. A medieval merchant would not have paid for some valuable modern resource like crude oil or lithium. But gold, on the other hand, was present throughout the Earth, and pretty much every ancient civilization valued it. Therefore, it was an accepted article of trade from Cipangu to the Cambemba.
Of course, silver was often used worldwide as a currency-like trade item alongside, and sometimes even replacing, gold. But silver's properties are just like gold in this regard, in its aesthetic appeal,
Ultimately, trade in gold in this way is more like to barter than to a proper currency. It is exchanged value for value with another trade good. It just happens to be generally accepted as a trade good anywhere.
An analogy of this within a solar system is finding some good that is valuable to anyone. In our solar system, anyone on Mars might value a comet full of water or ammonia. People on Earth might value rare Earth metals or liquefied hydrogen. People in the outer solar system would put a premium on solar energy, or, lacking that, food grown with solar energy. But to make a good that is really like a currency it should be a single product that is universally valued. Gold is still a viable candidate, given its continued rarity and appreciation as a marker of wealth and prestige. But a better candidate might be some rare element that is critical to some futuristic technology. Perhaps lithium for batteries, Helium-3 as fusion fuel, or some rare element (I vote Bismuth) as catalyst for some futuristic technology (Maglevs! Cold Fusion! Warp Drives!)
The Medieval Way: Send your brother
Lets say you are the richest man on Mars, and you want to get into the ammonia importing business for this nitrogen-stared world. Lo and behold, there are a bunch of ragtag colonists have placed a station in orbit of Charon which has cryovolcanoes blowing the stuff practically into orbit (Note: this is speculative!). However, your Hegemonic People's Republic of Mars credits are no good in the wild west of the Kuiper Belt. How are you going to establish a trading relationship out there?
Well, if you are anything like the Peruzzi, Fuggers, or Rothschilds, you will send a trusted family member to act as your factor out there. You see, you still have lots of valuable Martian industrial and agricultural produce that the barbarians of the Kuiper belt would pay good money for. So you sent your brother out there and every couple of months ship him all the goods that he could ask for. He will then sell the goods for whatever goods or currency is valuable locally and then when he has amassed enough capital, pay for a giant ammonia ball to be mined out of Charon's surface/atmosphere and shipped back to Mars.
Sending a family member or similarly trusted person is key, since the time gap for even light speed communications to Pluto is 4-7 hours (depending on orbital alignment). If your factor is cheating you, the local law enforcement won't care (remember, the Kuiper Belt is the wild west), and it will take months to get a replacement sent out there. Better to send someone you trust, and what is thicker than blood?
Once your factor is in place there, many other options become available. Wealthy persons at Pluto Station will be eager to do business with your factor, because they can obtain a letter of credit there that will be exchangeable on Mars for a whole variety of commodities. You see, the wealthy people of the outer solar system have the same problem that you do: their 'currency' might not be accepted in the inner planets. If they want something as pedestrian as old fashioned as a leather belt made from a real cow, where will they get the money to pay for it? This is where your factor comes in. They pay him in whatever currency is valid on Pluto and in exchange, he gives the a note that they can redeem on Mars for something that will hold value in the inner solar systems.
In actual history, this is how most modern financial institutions were born. The Peruzzi and Bardi of Florence used this method to begin the first international banking in the 14th century; the Rothschilds used this same methods to start the international bond market in the 19th. It would not be surprising if the interplanetary banking market started this same way.
The Modern Way: Central Government Currency
There is a long way to go before you can have a 'modern' currency such as that we are all used to with the Dollar, Pound or Euro (apologies to readers from other regions). When the US Government first issued the dollar, it was explicitly backed by 'reserves.' These reserves took the form of something with a value that people could trust, gold (see: The Ancient Way, above). This continued for a long time. The UK did not come off the gold standard until 1931; the US until 1971 (for international convertibility, for you pedants out there, which is what is relevant here).
A central government can still, even without the currency being officially recognized, issue a currency that is valid in other places. I have personal experience with the value of the not-officially-recognized US Dollar in Africa. The key is, you will have to negotiate each trade with an exchange rate built in (again, as I personally experienced, and African merchants are much more up to date on the exchange rates than I). If the central government is stable and trusted (as regards its debts and currency, at least), then any currency it issues would be valid. Given that the Republic encompassed tens of thousands of systems during the Star Wars prequels, for example, Watto would have been a fool not to have accepted Republic credits. He could have driven a hard bargain on exchange rate (he did have the only available hyperdrive) and then found another local moneychanger to exchange Republic credits for whatever he wanted. Unless the Republic had a 10,000 year history of defaulting on its debts, anything that is valuable somewhere else is valuable here too, minus the cost of transportation (which, incidentally, seems pretty cheap in Star Wars).
In this solar system, it would be reasonable for any large and stable governments, United Earth and the Hegemony of Mars, for example, to have currencies that are widely accepted. Even in the far reaches of the Kuiper Belt, where pirates are more common than lawmen, anyone could appreciate being paid in Martian Ducats, since they will be sure that they can get a good deal from someone else for that money.
This monetary system relies completely on trust in government. By trust, I mean that people trust that the government is valid. This has been the case for so long in Western nations that people just take it for granted, these days. But it wasn't so long ago, barely 100 years, that civil wars raged, governments were replaced, and hyperinflation destroyed the value of currencies. So long as the people of your solar system can trust in at least one government that is issuing its own currency, there can be an agreed upon medium of exchange.