Open source won the battle of software. The next battle will be about data itself.
It surely implies very hard problems when it comes to standardization and how data is exchanged. It might involve something like flatbuffers. How data is exchanged, what rights you have on it, how it is made secure, there are no ubiquitous idea or software that can reinvent the web because network programming is just hard and it wont change soon.
What I think could really be relevant is a database that syncs itself like the kinds of bittorrent sync and syncthing. Once you have atomic data that is spread across users, nobody needs to rent servers, and the data belongs to the users. That is a true and real way to reinvent the web, it also solves some of the controversial problems of the internet: advertising and surveillance.