Thanks everyone for your responses. I have looked at some of the links/videos and I think I will look into a "Wireless telegraph" design some more. I know the 4th grade kids learned about "wired" telegraph in school this year and it should be fairly simple to create a CW transmitter and receiver built with "nails on a board" and separated by a foot or two of space. This allows the kids to operate the telegraph key on one end and hear the beeps come out magically out of a speaker on the other side! Perhaps two separate boards can be built so they can be carried away as experimentation of transmitting over some distance.
The question now is: what is the absolutely simplest CW transmitter and receiver circuits? The frequency doesn't matter and the distance transmitted only needs to be about two feet. I can imagine the transmitter simply being an oscillator and a transistor for output power. The receiver will likely be a little more complex.
I suppose any form of crude "spark-gap" design is out since it has to work with a small battery and minimal parts.
A wired telegraph isn't a bad idea if that's also OK
http://www.w1tp.com/perbuild.htm Substitute a buzzer for the sounder for Morse as we know it (not just clicks).
One form of wireless which is simple is a ground communication system comprising a couple of audio amplifiers and ground stakes.
Spark gap is actually really simple provided you don't have to build the receiver. Just a relay will do connected to oscillate. Eg the relay part of this circuit
http://www.historywebsite.co.uk/Museum/Engineering/Electronics/history/sparktx.gif Add a transformer for a painful electronic shocker. Or attach it to
two ground probes to transmit through the ground. Use a crystal earpiece (hard to obtain) or audio amplifier for the receiver.
If you don't have to build the receiver that simplifies things a lot. You could actually have two old AM radios and use the local oscillator of one as a transmitter.
What you do is you tune one receiver to a weak station at the high end of the band. Then tune the other radio until you hear a beat note. Interrupt the battery
supply to transmit Morse. Super simple - just a bit of circuit board in the battery compartment might work - almost no construction.
Two $5 Chinese Pixie kits is probably about the simplest for tx/rx link. Otherwise a 1 transistor crystal oscillator on 3.58 MHz and another crystal oscillator used
as the local oscillator for a direct conversion receiver is very simple. Feeding in audio could make it AM.
It's a school project, not a space shuttle. Go for the simplest thing that works, will impress the teacher and other kids and get a good mark. Nothing more.