Why does "mining" Bitcoin have any value? Solving complex computer algorithms is actually a meaningless event. If you mine bauxite, you actually have bauxite. Mining Bitcoin you have... nothing.
With bitcoin, what you have is proof that you burned compute cycles. Yeah, I know, that's kind of weird. It's like if you mined, I dunno, dark matter and put it in a bottle. Nobody can see it, but yeah, it's there alright, and I can prove that my dark matter mining machine went through the proper mining procedures to gather this much dark matter. Or at least, somebody's dark matter mining machine did the work and I later traded them something of value for that jar of dark matter. Like I said, weird. But it's not precisely nothing.
Bitcoin (and similar schemes) has another thing going for it that no preceding electronic money system ever had, its developer(s) solved the scarcity problem. No previous e-curency scheme has had this one locked down. Even when it comes to "real" money, dollars and euros can be waved into existence without limit, but in an analogy to physical mining, there'll only ever be a certain amount of bitcoin.
A currency only works because something is behind it. That could be a commodity like gold or a vast military force. If there was a competition to solve calculus problems I'd do really well. I'd fail if it was based on baseball trivia. Why would we assign currency value to it?
Because we can trust that somebody burned compute cycles on it, and that the market'll never be flooded with wheelbarrows of freshly-printed bitcoin notes.
David, I know this sounds like I'm some kind of bitcoin proponent, and really I'm not. I don't own any, and never have. Frankly I don't understand it well enough to put any worthwhile amounts of money into it. I just think that it's interesting that this shadowy person or group of people have put together something that's in many ways more open and honest than "respectable" central bank currency.
And yeah, it looks awfully bubble-ly to me too. Gold mining stocks are probably less risky. At least you have something you can point to and say "See, I own x% of that mine over there."