@ryanmw2002:
Placing an RFID/NFC reader below the board and placing passive RFID tags on the objects will work just fine for both proximity and identification. Certain models of reader (with their corresponding compatible tags) allow a range as much as several cm.
You might want to tweak the vertical positioning of the reader if you want to mark the detection exactly when the object actually touches the surface (as opposed to 1-2 cm above the surface).
The RFID reader cost might be a little high (> 30 USD) if you buy from a hobbyist shop or in small quantities, but you can very likely get the price < 20 USD if you buy from manufacturer.
What are you trying to design, i.e., what's the application?
Look through the following pages to get some perspective: