Why I can't use my old phones to do this is beyond me. I have one windows phone lying around and a few android phones. Everything is there in a neat package but the manufacturer make it so hard to tinker with them.
What's preventing you from using an old phone for this? There's no specific hardware (other than driving the display, which your phone already has), right?
Blame the rise of the app store. The old Windows Mobile phones were fully open and work just fine for what you want; write your app, load it onto the phone, and just launch it.
Once did a project like this that grabbed Seattle's realtime data for some bus stops around my place. It was an old Windows CE PocketPC thingy, so the tools were primitive (.NET compact framework 1.0, basically).
Maybe a bit too primitive. It worked pretty well for about a year, until I found that the CompactFlash WiFi adapter was only 802.11b and there was nothing newer available that was compatibility with the PocketPC. It ended up being a choice between keeping the gadget running, or being able to run my WiFi network at 802.11g speeds with modern (WPA) security, so the latter won out.
For a one off project, why wouldn't an old android phone work? It isn't that hard to sideload an app onto the device, and apps can be developed to comm with external devices through USB OTG adapters.