It can actually happen. You can get to a point in freelance Web writing where you don't have to actively search for work. It took awhile, but I am so grateful that I don't have to search for a couple of hours every day before I start the actual work.
How it Happens:
The first step is to have a visible Web presence. I have a writing sample site, Squidoo lenses and a couple of blogs to keep my name out there. Clients write to me through the sample site or the lenses and ask me about rates and tell me about their projects. Most of them then disappear, perhaps believing me to have too high a price, perhaps simply capturing my email for the purposes of evil. But, some offer little projects that I can do without having to search out the gigs.
I also frequent a fabulous writing forum full of women writers. I regularly do projects for some of those ladies and have had nothing but good experience with them.
I have a profile on Elance full of good feedback and a portfolio full of writing samples. I get invitations for projects through Elance as well as scoring regular clients through the site. Many of my regulars are people that I first worked with through Elance.
I write directly for a company. I've been writing for a specific content company for almost a year. That work is there all the time and brings in a steady check. I'd like a second company to contract with so that I have two meaty checks every month instead of a bunch of small ones. Perhaps soon.
I write directly for websites. Suite101, AC and others pay me residuals that add up to a small passive income. If I wrote more for them, I'm sure it would be a more impressive amount. My payments are small, but it's a good revenue stream.
I make a little from this 'lil blog. I haven't done it as much lately, but I've made a little money by writing sponsored posts for various companies. There are people out there who do quite well by blogging and using paid posts.
When self employed, having different income streams is incredibly important. If you don't have a boss who will write you a check for the same amount every month, you can't rely on one single source of freelance income. Keep the work diversified and keep your name out there and there will come a time when you no longer have to hit freelance job sites.