Wow, that would be neat.
Reversi is a two player game. What you do depends on whether there is going to be two players (possibly hard) or one playing the Android (very hard).
You need a game board. You can make that with a Table Layout and Buttons or you could use the Canvas with hot spots or possibly using sprites as the playing tokens. This is easy so far.
image is from WikiPedia http://en.wikipedia.org/wiki/Reversi
The following AI2 games have possible algorithms you might need
This web is for programming with Object Pascal, however many of the techniques might be directly compatible with AI event driven programming
http://www.delphiforfun.org/Programs/delphi_techniques/gameboards_reversi.htm I would use this as a guide. The TPanel example would be analogous to using a Table Layout with buttons. The Tstring example would/should work using the 36 buttons programming example where you would increase the buttons from 36 to 64. Either one of these techniques might work using App Inventor.
Do some reading. Decide on a strategy and try some blocks. Let us know how this works for you and then show us what you have done.
Good luck,
Regards,
Steve