Create custom PCBs with circular contact pads on them (like a bulls-eye) to put on each square of your game board, and your pieces would have unique contact points on their bases. Or unique resistance value(s).
ATTinys are pretty cheap and could be programmed to report an identifying value when queried (via serial, SPI, infrared?). Again your board could supply power to the pieces and just a single digital output to each game square would be all you'd need to trigger the piece to report its ID along a common bus. That should be doable for about $2 per game board square and $2 per game piece.