Thanks for some very interesting answers. I'm trying to write some followup questions to some of the topics that have been brought up.
I have put together some scenarios, or use-cases. I'd love some thoughts about these too.

1) I am a conservative senator with very traditional values, focus on the family and such. I do however in my spare time purchase gay porn. Naturally I don't want to associate my visa-card with such activities, so I buy some bitcoins with said card and then spend those in a shop which caters my interest. I do this though TOR or some other anonymous service, perhaps a VPN bought with bitcoins. How long, if ever, does it take for the journalists to show up at my door?

2) I am a computer professional who decide to run a service in I2P that is somewhat "gray" in the legal sense. A service that has both legal and illegal uses. I allow people who wish do donate bitcoins to this service. I already have a bitcoin wallet which I use for things that are 100% legit, so I use a receieve adress which is only used in the I2P-service. Will my anonymity be compromised by recieving bitcoins? Will it be compromised when I spend coins on the "open" internet?

If these questions have been answered a few hundred times, perhaps even in the FAQ I apologize and will hang my head in shame and go read those, if you'll just give me the link. I haven't seen anything about this, but perhaps I haven't looked hard enough.

I'll see if I can't come up with a few more scenarios later, if I can think of any.

Thank you for your answers. I still don't quite get it though.
Ok, so bitcoin isn't just a payment method, it's a currency. And it doesn't have a central bank, the bank is seen as a problem here.
That would draw some libertarians, not many others.
And some technocrats would be interested in the implementation. For me, I don't see any big benefit yet.

The way I  see it.
Fees, too small to warrant switching form a well established currency.
Chargebacks, solveable by insurance.
Freezing, a problem if you're wikileaks, not for many others. Don't keep all money in one place and you should be fine.

Here's where I see some benefit.
Full anonymity is possible if you're only doing virtual things?  But not if I mix the two. When I buy my first pizza for my bitcoins they can link that to my I2P-website.
Is there somewhere I can read about how to be fully anonymous? I'm thinking of ways to reward those who provide valuable services to I2P, nothing illegal. Just to be clear about that. Some things might be questionable though. Supporting a I2P-torrenttracker might be one of those.
However the person running such a service has no real use of the coins, other than pushing it around the I2P network. Right?

For regular services on the internet I would not accept Bitcoin. Too insecure. Not possible to insure against value fluctuations. Among other things.
It does have some benefits, but it's no "killer app".
Find some place where bitcoin "fits" and build a community there. Perhaps the anonymous networks, perhaps somewhere else. I'm afraid bitcoin will never take off, just existing as a fun thing for the libertarians and anarchists. :-)
I would love to be wrong though.

Hi.
Just curious about what problem bitcoin solves.

If I should setup a service in I2P and have people pay to use it, bitcoin isn't anonymous enough.
As soon as I use any of my I2P-bitcoins outside of I2P, I'm outed.
And anyone who wants to buy things from me would have a paper-trail linking their purchase from me to their regular account.

And if I setup a service on the regular internet, why should I bother with bitcoin and not just use VISA/Paypal/anything-else.

I'm trying to see a justification for bitcoin, but I just don't see it. Please educate me.

Tim