You could try a Nook Simple Touch because they are old tech, so available for next to nothing, you can readily root them to run your software ... and the serial port is well documented so you can attach sensors. See one I've converted into a flight computer by attaching a GPS ..
https://vimeo.com/67164582 for an ancient video of it in action. Good luck!
Obviously the hardware hacking requires some skill in soldering (not much, I'm useless!) but you may not need that if you just want to run an android weather app permanently on the device.
Otherwise, if you're in Europe just buy a device with Meteotime transmission receiving capability. Good weather forecast no matter where you are, and no internet connection required.