There are two ways the future of computing can go, in my opinion, and we'll probably get both of them. First, the network starts blurring the lines between what is local and what is remote, to the point where you'd talk more about your services and data than your computer. Wherever you go, as long as there is a network connection, you can get to your own files, contacts and services, because computing is such a part of our lives that it's woven into the very walls of our existence. Every surface is a computer.
The other direction is to gain more power in our phones, up to the point where they are our primary computing devices, capable of all our everyday actions like playing, reading, creating, communicating and working. Meshed with local services like big touch screens and cameras, they provide everything we ever need, right in our pockets. Everywhere I go, I have a computer with me.
It's a subtle difference, but quite important when the network goes down. If you have a local copy of your files in your pocket, plus the apps to use them, you can still work, to an extent. If everything you have is online, including your files, then you have nothing until you can get a connection. In big modern cities, that might not be a big deal, but travelling between them, or to an area dominated by a rival network provider, you'll be out of luck. In that way, offline storage and power is more important than online, because you just can't beat the availability.
Mokalus of Borg
PS - The network is a lot less available than you think.
PPS - This is why it worries me when everything is online.