Alas, experience suggests that most people don't value social networks enough to pay for them. Moreover, once a social network has locked in a user and their circle of friends, it's easy to discount their utility as a customer (they're locked in: where are they going to go?) and try to monetize them in other ways.
But I think examples like Sparrow (Mail app) represent the idea that if you make it obvious that it's worth paying for, people will do it.
Maybe if they had a freemium system. It starts off free but if you want no ads and more features you pay $X/month. A user could lobby friends to make the jump to the paid version.
It could simply be, that it was too early for LiveJournal? Back then – admittedly, now in many ways – the idea of a social network was new. It's value had yet to be proven at that time. However, with almost 1 Billion people on Facebook the value of such a network has been proven.
However, while I say 'I want to pay for Facebook' I'm really saying I want to pay for a service that allows me to keep in touch with my friends and nothing else. Not a service that allows me to keep in touch with my friends while also selling my data, and the problems that go with that.
I personally think that Microsoft should either buy or merge with Facebook, wipe the slate clean, redesign it Metro UI style, and charge for it. No ads, just a better overall service. I wouldn't hate Facebook so much if it were actually fun to use.
Either way, I feel they will always have a conflicted product vision so long as their users and customers are separate entities. Imagine a feature that's brilliant for the user, but terrible for advertisers? Well, hard luck, it's off limits. It's strange that one party is using and another is paying. It profoundly limits the scope of the product.
Well if any major tech company would go after them it would be Microsoft. Facebook's integration in Bing, Microsoft buying Skype then integrating it with Facebook, etc.
I agree though. The first thing to do would be to get rid of the advertisers and charge for it.
Microsoft Research has so.cl (that's the URL), which integrates with Facebook and adds some "social search" and video chat features. It's not competing, just experimenting with different functionality.
(Historical example: Livejournal.)