Update: Based on some helpful comments on /r/bitcoin, I edited my original post to clarify that Bitcoin derives its value, only in part, from the costs required to produce it. However, without that it would be valueless, even if there are other things that contribute to its value. Not sufficient, but necessary. Some fellow engineers I work with have been mining and trading Bitcoin since well before the mainstream hype of the last few months, and in talking to them I've become increasingly interested in it as well. It is one of the more elegant technological ideas to come along in a long time, and its greater economic, sociological, and political implications are also fascinating to me.
But when I first heard about it, I was hesitant to treat it seriously based on one fundamental doubt: how could a bunch of numbers spit out by a computer have intrinsic value in the same way that gold can? I understood how Bitcoin could have extrinsic value, based on things like trust and hype, but if that were all it were based on, why would Bitcoin be worth more than any other arbitrary currency one could create out of thin air?
And then I spent some time learning exactly how Bitcoin mining works, and discovered that there is, in fact, intrinsic value to the currency. (Funny how ignorance can lead you to dismiss things like that!) In order for Bitcoins to be created, a computer must solve a difficult math problem by guessing a number by brute force. This requires a running computer (these days, a powerful computer specifically built for this type of math problem), which in turn requires electricity, which was probably made with a fossil fuel or nuclear generator. So, in a way, you could say that the value of Bitcoins is at least partially derived from the fuel used to create the energy needed to power the computers that mine them.
But, you ask, what happens as computers get more and more powerful and energy efficient? Shouldn't Bitcoins get easier and easier to mine, dropping the amount of energy required to mine them, thereby decreasing their intrinsic value? Turns out that part of the ingenious and elegant design of Bitcoin prevents this from happening. The difficulty of the math problem that the mining machines have to solve changes dynamically over time. The system as a whole aims to stabilize the difficulty such that these math problems can only be solved roughly once every 10 minutes. If the computers start to solve the problems faster, the difficulty across the system is increased. If the computers start solving the problems slower, the difficulty is decreased.
To be sure, there are other factors that contribute to Bitcoin's value other than trust and hype. It shares many common characteristics with gold: durability, divisibility, combinability, homogeneity, and scarcity. All of these things factor together, along with the sociological stuff, to give Bitcoin its total value. But if it were possible to mine Bitcoins without expending resources, I believe their value would fall to zero. (There is another stopgap against this built into the technology: the total number of Bitcoins is capped at 21 million, so even if down the road it were theoretically possible to mine Bitcoins for free, only up to 21 million total could be harvested. That hard cap also contributes to Bitcoin's scarcity, and therefore its value.)
And so, it is this fixed degree of difficulty, inherent in every single Bitcoin that will ever be mined, that ensures that there will always be some level of effort required, and therefore some baseline value in the coins. Without this fixed difficulty, computers would be able to simply pluck Bitcoins out of thin air, and despite all the other valuable characteristics of the currency, it would in all likelihood be worth nothing.