If you don't mind being joined to Google at the hip, there's a great Linux-based cloud desktop operating system called Chrome OS. It's on Chromebooks. If, however you want a Linux-based cloud desktop operating system where you can rely on non-Google cloud apps, you have another excellent choice: Peppermint.
Specifically, the latest Peppermint is based on Ubuntu 14.04.2 point release, which includes the 3.16 kernel and an updated graphics stack. Where Peppermint steers away from Ubuntu, and other major Ubuntu-based Linux distributions such as Mint, is its reliance on cloud-based applications instead of traditional desktop applications.
So, for example, instead of using LibreOffice or the default office suite, Peppermint encourages you to use Site Specific Browsers (SSBs) for office applications. With Peppermint's Ice application you can easily set up cloud-based services, such as Google Docs or Zoho Docs, to act as your "desktop" office suite.
You can do something like this with a web browser's "Create Application Shortcuts" tool, but these don't integrate well with a Linux desktop menu. In addition, it's hard to get an application shortcut out of a menu system once it's there. Ice makes integrating web apps onto the desktop easy.
SSBs also allow applications to function in a more standalone method rather than running them with all the overhead of a full web-browser.
You can, of course, use the usual Ubuntu software installations tools, such as the Ubuntu Software Manager, to install the standard Linux desktop apps. That would be missing Peppermint's point. Peppermint delivers a fast Linux desktop, based on Lightweight X11 Desktop Environment (LXDE), that's meant to run cloud apps well on minimal hardware with a good Internet connection.
It does a great job of it on my Lenovo ThinkPad T520 laptop. On this 2012 business laptop with its 2.5GHz Intel Core i5 Processor, 4GBs of RAM, a 500GB hard drive and an integrated Intel HD Graphics 3000 processor, Peppermint ran like a champ.
Peppermint's developers claim it will rum on pretty much any x86-based architecture with 512MBS of RAM. I think they're right.
Besides using LXDE and cloud applications, Peppermint borrows the best features from other Debian-Ubuntu-based Linux distribution. For instance, it uses Linux Mint's Nemo file manager. What I like about Nemo is that it makes it easy to mount and manage a wide variety of remote network shares such as Windows SMB shares, WebDAV, and SFTP over SSH. All of which, I might add, I use almost every day on my desktops.
For a web browser, and its SSBs, Peppermint uses Chromium, Google's open-source version of Chrome. You can also install and use Chrome. What you can't do is is use Firefox for SSBs. That's because Mozilla stopped working on the project, Prism, which enabled desktop Web application functionality.
While Linux Mint will continue to be my favorite desktop of all, and I'm a huge fan of Chrome OS on Chromebooks, I'll be installing Peppermint on older systems from here on out. Its speed on minimally powered PCs and cloud-based applications makes it an ideal Linux for low-powered PCs.