Although there have been similar baked schemes, they all still used TCP/IP and other Internet staples. What I propose is a radical shift into a much more decentralized Internet.
First, bandwidth. Just like on a filesharing P2P network, you share your bandwidth to perpetuate the network. All of your
unused bandwidth is efficiently used to pass files onto other peers.
The system, much like one called ANts, would be completely anonymous due to the vague addressing scheme. Likewise, domains would not exist and you would request sites based on a self-assigned name (much like a filename on a computer file).
Although this means that more than one entity could share the same name, and allow for impersonation, it should be perfectly fine: sites could also be given a randomly generated ID number to stop unintentional imitation (browsers would separate search results into groups based on their ID). And the network architecture itself will help curb intentional impersonation: only popular sites will be widely perpetrated through the network.
Wireless would be the easiest choice for this network, because running several wires to nearby physical peers would be very expensive. With wireless, every wireless ethernet card could be a peer, allowing peers to easily connect to each other. And for connecting between major centers, the software can be designed to keep acting as passive forwarding peer even when the laptop or other device is off. This way, if there's enough traffic on the interconnecting roads, the Internet will leap across the highways like magic.
Much like the BitTorrent network, every page you download is stored in a cache until you have sent it to a perr a certain number of times, settable by you. Two or three seems like a good default.
So, when you want to create or update your website, you simply send it to your closest peers with a standard protocol intruction to proliferate it. You could add a life parameter to this proliferation, telling it to proliferate X number of times before stopping. Searches and many other commands work in the same way. Data may be encrypted at its start and end points for added security.
Overall, this network has many advantages over the existing one. Paid-for domains are made unnecessary, as is webhosting, resulting in a truly free Internet. No longer can any material be repressed without major consensus of nearby peers. Big corporations have no more power on the Internet than any individuals, and with complete anonymity, ISP subpoenas go right out the window. In fact, there isn't even an ISP to begin with. No AOL, no MSN, no Comcast... it's a paradise. And with no central authorities and not even a central server or path for any one site, it is much more reliable.
If the vast technological challenge of making this system fast enough for everyday use can be overcome, I think it would be a radical tool of intellectual freedom and of even greater utility than the Internet of today.