Cool idea.
This just popped into my head: Hallmark Cards, at least at one point in the past, sold "recordable" greeting cards. The person buying one could take it home, record an audio greeting, then send or hand the card off as they would a normal card. Upon receiving it the person who opened this special card would hear the greeting.
If you could find one of these, and take it apart, I'm sure that the electronics inside would accomplish exactly what you're looking to do. It would be at minimal cost, and to customize it for your layout you could add an extension pushbutton to activate it remotely, in place of its existing card-open/close switch.