Short answer: Yes.
Longer answer: Yes, they are money, because they can be used for paying and storing value.
Long answer:
To really understand why crypto currencies are money, we need to know what money is. This question is dicussed in greater detail in another posting. Here is a very short definition: Money is whatever can be used for paying and for storing value. As crypto currencies do have these properties, they are money. So, actually we should probably rather discuss the following question:
Why is this question interesting anyway?
While I find it hard to doubt that crypto currencies are money, many seem to find it hard to believe that they are money. Many people would, perhaps, admit that crypto currencies are money "in a certain way", but they would still distinguish between crypto currencies and "real money". I think it is interesting to discuss where these doubts come from.
Why should crypto currencies not be money?
I think the following issues might make some people think that crypto currencies are not money:
- Crypto currencies are not yet very practical.
- Crypto currencies are not issued by states.
- Crypto currencies are somehow abstract, complicated and technical.
- Crypto currencies are "fiat".
- Crypto currencies are "speculative".
As concerns the first objection: If our definition of money makes sense, then "not yet very practical" could either mean that you can not yet use crypto currencies very well for paying, or that you can not yet really use them for storing value, or both.
Currently there are very few places where you can pay with crypto currencies. Besides there are many different crypto currencies, so it is hard to know which one to buy in order to be able to use it for paying. And in most cases you can also use "normal money" instead of crypto money. In most cases there is no real reason to use crypto money. So, paying with crypto currencies is possible, but still quite difficult. But this is not an argument against crypto currencies being money. In their early days electric cars were a lot less practical than they are now. And fewer people used them. But that didn't prevent them from being cars.
When it comes to storing value, in a way crypto currencies, or at least certain crypto currencies, were better than anything else in the last years. But most people would expect a truly value-storing currency to be rather stable. Again, crypto currencies are quite new and therefore their functionality is still being developed. That doesn't prevent them from being money. Besides, many fiat currencies, such as the Venezuelan bolivar or the Argentinean peso are also not stable at all. But noone doubts that they are money.
I think objection number 2 is the one that convinces most people. Many people seem to think that money is something that is intrinsically related to the state. Maybe that is because that which money is used for, paying, has legal implications. And anything legal seems to be related to the state by definition. However, while the notion of "legal tender" does exist, first, there is no real necessity for the notion of legal tender to exist and secondly, there is no reason why crypto currencies could not be declared legal tender. Actually, in El Salvador and in the Central African Republic Bitcoin has been declared legal tender. (A currency is called "legal tender", if payments in this currency legally extinguish debt.) If John and Mary have a contract stating that John owes 10 Moneros to Mary, then John's paying the 10 Moneros to Mary would extinguish this debt. For this to work there is no need to declare Monero legal tender first or to convert the 10 Moneros to euros, dollars, or any other fiat money in order for John to be able to use whatever is "legal tender" in John and Mary's country. Why should John and Mary live in the same country, anyway? What if John lives in Canada and Mary in Venezuela? Which jurisdiction applies? Which state gets to meddle with their business and force them to accept something as legal tender?
I think paying is something that exists (logically and historically) before and outside the state. When there is no state, e.g. after a complete breakdown of public order, there is still business. People still buy and sell goods and services. And they use as money whatever they see fit. After Germany's complete breakdown at the end of World War II cigarettes were used as money. There was no real public order, so nobody could prevent people from using cigarettes as money.
I would say that there is no need for money to be in any way related to the state. Money is whatever is used as money. The state has usurped the notion of money by claiming that nothing is money unless it says so. However, with globalization it is more and more difficult for states to control everything. And there is no real reason why only fiat money, i.e. state-controlled money should be used for paying.
As concerns objection number 3, I think one important aspect of crypto currencies is that you can't touch them. You cannot hand someone a Monero like you can hand someone a dollar. Also, using crypto currencies necessarily involves complicated technology. New technology may have unknown problems and it may also change. And not everyone may have access to the necessary technologies. However, the same can be said about many payments with fiat money. A lot of complicated technology is involved in traditional money transfers. Although this technology is older and might therefore be assumed to be proven, it is rather intransparent compared to the open source crypto technologies. I would say the fact that crypto currencies involve more technology is no reason to think they are not money.
Oddly enough, many people seem to critize crypto currencies, as opposed to "normal money", on the grounds that crypto currencies are "fiat". Usually the word "fiat" is not used in those criticisms, but the idea stated is:
Crypto currencies have no intrinsic value. They are just data and algorithms. As soon as we stop believing in a crypto currency, it loses value or disappears altogether.
This is obviously true. But the same can be said about "normal money" such as dollars, euros, pounds, and so on. "Normal money" is fiat money, i.e. it has no intrinsic value. It has value only as long as people believe that it has value. The only difference is that states are responsible for fiat money. And many people seem to trust the state. However, there are obviously also very untrustworthy states, especially when it comes to money. And you never know when a trustworthy state starts becoming untrustworthy. Besides, even if a state tries to protect the value of its currency: If the market disagrees, then the currency will go down. Or it might even go up, such as the Swiss franc, despite Switzerland's measures to weaken it. The prices of things, and that includes currencies, are determined by the market. The state can invent prices, but the market does not have to believe in them.
As concerns objection number 6: Yes, crypto currencies are still very "speculative". However, every good, and that includes every type of money, is speculative. We do not know the future of anything. All we can do is speculate.
There might be issues that I have overlooked. But I cannot find a reason for not believing that crypto currencies are money.