You can buy a relatively cheap (<$5) RF Transmitter & Receiver kit for Arduino.
I would imagine that a solution could be as follow: low cost *duino + temperature sensor + transmitter > receiver + *duino + display.
This solution may not be cost effective when compared to a commercial wireless weather station that you can buy from Walmart. Of course, buying some pre-made modules hardly classifies as building something from scratch, right. Yes, you may learn something, of course, but there are more interesting projects.