If you want something ubercheap just buy a used computer off Ebay or Craigslist. You can get a reasonably equipped machine with a LCD for around $400 nowadays, so if you want something cheaper then that you’ll have to go used. A cheap laptop can be had for $200-300 as well that’ll run Windows XP just fine.
And you need a hard drive, but hard drives are incredibly cheap.
Alot of the work has already been done for you. You are describing a PDA with a big screen. PDAs normally have processors that are 150 something mhz, flash memory only (32~64 mb).
I’ve pasted a link here to the Q1 for everyone’s info. The Q1 seems like it does a lot more than just the internet though.
Other things to think of or research are thin clients. These are terminals with no CPU or memory, they instead use a central computer. You can find these in libraries used as catalogue access computers. Also, look up “small form factor” PC. Basically a Mac Mini.
It will take alot of legwork to figure out what would be cheapest. Components that are cheaper may require more programming to run your OS. I’m assuming that you would have a proprietary OS. Since you only have one application, it would make sense to not buy licenses from Microsoft for Windows or Windows for PDAs.