Back in the day, before the dot-bomb, there was a site that did something similar. A few sites, actually. I think one of them was iWon (
http://en.wikipedia.org/wiki/IWon), which offered points for performing certain actions on various websites. Others followed a treasure hunting model. You had to find certain pieces of content, or perform certain actions, to get points. They monetized by signing up partners (since this generated visits) and advertising.
That being said, I think this is an interesting idea. With Foursquare, Gowalla, and others having popularized the "check-in" idea, the market could be ready for this kind of a service again. If you can build a system to prevent gaming, you'd have a good, defensible product on your hands.
I think it's definitely worth a try. You can even scoop up some of the previous ideas (treasure hunt, rewards for actions, etc) and incorporate them too.
Good luck!
[Edited & added some more info below]
Ah, I just recalled a few other attributes of the iWon system. They started with a portal page and rewarded you for clicking on certain links. This obviously is different than visiting a website whenever you wanted and checking in - and I think the latter model is better.
They also had a time restriction. You could only perform a certain number of clicks a day, and perhaps even a certain number for hour. Those may not stop people who write scripts to game your system, but it's an element you can consider.
I'm guessing you've already considered the social aspects of this too, yea? For example, check-ins could appear on Facebook, Twitter, etc. If your system included a friends concept, I could see which websites my friends like to visit, and perhaps go there too.
BTW, if you can pull this off, you'll also end up with a great database of user behavior. You'll know who the users are (Since they signed up on your site), you'll know which offers & sites they like to visit, and you'll know how frequently they do so. Not that I'm highly recommending this, but such a list of users could be very valuable to marketers.
As the website owner, I'd love to be able to contact my visitors. Some sites may want to offer visitors free content, free stuff, easter eggs, etc. As a marketer, I'd love to know who visits my competitors, or related sites, so I could offer them incentives to use my service instead.
P.S. I wonder who'd become the mayor of Hacker News ;-)