I enjoy having a weather station at home. It is hooked up to APRS, weatherunderground.com, and I even have a weather webpage. One of the standard exchanges of information in most general QSOs is the weather: temperature, rain, …. I also like telling the folks in Florida that my humidity is 40% (I am not a fan of humidity having expierenced Fort Benning, GA in the summertime and monsoon season in Korea, not to mention my unairconditioned room at The Citadel (although I hear they have air conditioning now!)). 
It is easy to look at my desktop display and get all the data I need. I have heard of some folks who have a way to pull their weather data directly from their weather stations and input it into their PSK QSOs. Pretty slick, but I have never figured out how to do that (… yet).
All that being said, I do not get into weather prediction that much. If I see the barometer dropping, I may check the locak National Weather Service radar to see if anything is moving in (weather here moves from west to east). But if I wanted to get into weather prediction, this would make an interesting homebrew project: The Tempest Prognosticator.
Developed in the 1850s by Dr. George Merryweather, this device used leeches that would ring a bell if a storm was approaching. The device was even featured in Britian’s Great Exhibition of 1851. Despite the publicity, Dr. Merryweather was never able to get the government interested in putting the device into use.
I am sure there would be a way to interface the slugs with some kinda of Arduino device that would send out weather predicitions via APRS data. 🙂