Bitcoin only has intrinsic value because you expect it to continue to be useful as a medium; in order to use it as a medium you need to get some temporarily; and there's only so much of it.
You could think of Bitcoins as representing some percentage of the total amount of theoretical bandwidth of the Bitcoin blockchain security system. If you have .01% of that total functionality, and it's useful and effectively irreplaceable (which of course it isn't because of other similar currencies, in my opinion), then that's where the basic monetary value of Bitcoin derives.
However, because it can be valuable even in incredibly tiny slices, there really isn't much intrinsic value besides the expectation that in the future people will consider it to have value for rational and irrational reasons.
There's no analog to future expected earnings/dividends of a stock... It's much more like gold, where there is some slight industrial value but mostly you just expect its various entrenched forms of psychological value to persist. Because you can replicate Bitcoin and other cryptocurrencies relatively easily, it's not hard to imagine Bitcoin as a specific currency/system becoming obsolete or eclipsed.
Its value only really matters because having some value means there's an incentive for people with access to energy (whether they are paying for that energy or overusing their college unmetered energy or stealing it) to use that energy to perform math that is useful to making Bitcoin work.
Why mining works
Basically you can enter any transaction to any local copy of the Bitcoin ledger and wait for that transaction to propogate throughout the system. But if you enter that transaction without actually encoding it using the secret key that only the previous owner of that portion of Bitcoin knows, your transaction entry will fail to be confirmed by a bunch of math.
If you do really get the key from the previous owner, and now you're the owner, that transaction will succeed in being confirmed by a bunch of math. To keep people from being able to hack this just trying lots of different keys until one works, it has to take a shitload of time and energy to confirm a real transaction. So Bitcoin wouldn't work if people weren't willing to perform that shitload of time and energy.
It's very important to the success of Bitcoin specifically that it continue to be perceived as the frontrunner in terms of adoption and support within various apis and real life businesses. If that decreases, its value as a protocol and as a psychologically recognized currency decreases, which means eagerness to mine will decrease, perhaps in a downward spiral.
My guess is that this will happen at some point and that's why I don't own Bitcoin, FYI. All the reasons why Bitcoin is here to stay are reasons why some Bitcoin replacement, or many of them, will also or instead be here to stay.
The 51% problem
I don't understand this well, but I gather that the "51% problem" is the observation that if you controlled most of the computers that confirm the math, you could just be like "Yep, Warren Buffet sure did just give all his Bitcoin to Ben Wheeler, I totes confirmed the math guys!"
This would work because really the only way other people can be sure is if a) they confirm the math themselves, which you can't keep doing forever or b) at some point they just believe you because enough people/machines say the math is confirmed.
The blockchain ledger as store of truth
We normally think of Bitcoin values as being valuable because you have a certain amount of them and can spend those.
But the functional value of Bitcoin doesn't require you to have more than a minimal amount of the blockchain addressable number space in use. Even a tiny amount of Bitcoin, like .0001 BTC, gets the benefit of full blockchain confirmation of current ownership, reliable within minutes.
In that sense it's sort of like how businesses use the classifieds to make public announcements that are widely confirmable. You can do that in a tiny amount of newspaper space -- the value is basically your ability to do that at all, and be able to show it in court if you ever need to to prove you were being transparent about something.
Lots of startup technologies need to disseminate secure and fast updates about who owns what -- "I'm committed to paying you $50 to use that conference room from noon to 1pm". That's currently a headache to administer because annoying users dispute credit card charges, the server needs to worry about uptime and speed so that all users see that the room is reserved, and security is a headache because someone could charge your credit card erroneously or impersonate you and appear to make lots of reservations.
So what if we treat tiny bits of .0001 btc as contracts -- you enter a contract with me, on some form, and we confirm that by transferring a particular .0001 btc that represents the ownership we are transferring. There can be no doubt that you now are the owner of that good, whatever it represents.
It's not hard to see like 10 problems with this around the edges, but the core insight is that we now have this technology that lets us agree worldwide on who owns something. If we stop thinking about it like money, it's something that could open up property value in all sorts of new ways.
Think about the economics and political thinking of neoliberal folks like de Soto (see http://en.wikipedia.org/wiki/Hernando_de_Soto_Polar#Main_thesis ). Not everyone thinks they're right, but universally agreed ownership contracts could address many of the issues of shaky property rights in politically unstable places.
They could also mean transferability of assets in a peer to peer manner.
As I said, I'm only starting to vaguely understand this... the journalism is lagging the technology by a big distance.