Asus based their original Eee PC on linux. Later models had bigger screens and were Windows based but you could install Linux if you were that way inclined. All a bit dated now, there are better netbooks around, some with Linux pre-loaded.
I have a tablet, and bought a protective cover with built-in USB keyboard. Assembled, it is effectively a netbook.
Google's Android OS is a flavour of Linux, and is far more popular for mobile devices than the GNU/Linux most people think of as Linux (in fact it's way the most popular mobile OS in the world today). But beware old versions of Android, they were designed mainly for phones and their handling of text documents such as long emails sucks worse than an Eee PC with Windows. The latest release, "Jelly Bean", is great but only slowly working its way onto the market.
Top of my Christmas list is the Asus Nexus 7, a 7" tablet with Jelly Bean, that I can swap for the old tablet to create one cutting-edge netbook and a digital photo frame.
Or, consider the Samsung Chromebook. This uses Google's other OS, ChromeOS, and with its bigger screen is perhaps closer to a lightweight laptop. It keeps all your apps and stuff in the cloud so is physically simple and cheap. When connected via fast broadband it's said to be fantastic, the future arrived today, but when it loses its connection it gets badly crippled.
"Klinger, do you know how many zoots were killed to make that one suit?" — BJ Hunnicutt