How to build cheap radio propagation beacon?
Honestly, there isn't a lot of work to sending Morse Code. All you need is a frequency-stable oscillator that you can turn on and off.

The closest I can find to a "single chip" solution is one of these oscillators. Add in some low-pass filtering and you're there...

The way I'd expect to make a circuit is to use a discrete crystal oscillator (like something you'd find in the ARRL handbook) and add another transistor to be able to switch the oscillator on and off. A simple Morse Code radio is a good place to start. 

A straight key can be replaced with a transistor, and that transistor can be driven by your Arduino.