Nintendo is usually pretty resistant about opening their hardware to third party development (maybe I am wrong with the DS), so anything like this would probably have to be a hack.
The processor probably isn’t strong rough to do much more than the most basic web surfing, even if you could hack it.
@J0E is right, for what you’re looking for, a netbook makes more sense. Its only about $100 more expensive too, which isn’t that much when you consider the price of video games.
Yeah, if you just want something small with an internet browser, you can buy a 7” EeePC for like $200 US. You obviously won’t be able to play DS games on it but you can use regular PC applications which would probably be more valuable to you.
it should have a browser for the net… aside from that you could try to find some sort of “homebrew” application. You would need a flashcart that works on the DSi… the only one I know of at the moment is the Acekard 2i