I don't know of a browser that does that, but as a loyal user of the browser Firefox I can recommend an extension called StunbleUpon. After a short, free registration you can customize it to randomly choose websites that others have rated highly from the couple dozen categories that you select from. All you have to do is click the "Stumble" button on the StumbleUpon toolbar and it will randomly select and load a new website. You can also tell it to choose a website at random in a specific category and then rank the page. If you're interested in finding new websites based on your interests, that is the perfect extension for you.
In regards to Firefox, it is a free alternative browser that's more secure than Internet Explorer and has hundreds of extensions (add-ons) that can be downloaded, just like StumbleUpon. I highly recommend giving it a shot.
Hope this helps,
I'm and old cripple with nothing to do but play on the net.
Several years ago I found a browser or something that would allow me to open a site at random. I can't remember anymore about it, except when I hit next I would get a new random site to view.
I lost eveything when my CPU froze and when the repair shop fixed it I had lost that program.
Any one ever heard of something like that.