Tom -
While Googling our pet problem I came across this piece of advice, which worked for me - hoooooray!!
From http://apple.stackexchange.com/questions/146644/airdrop-bluetooth-or-wifi
| There's a way to use Airdrop without Wi-fi connection There's a way to bring back the "Airdrop" to Finder again even that you just in Ethernet/LAN connection. Simply run the command terminal below: defaults write com.apple.NetworkBrowser BrowseAllInterfaces 1
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And, from http://www.peachpit.com/articles/article.aspx?p=2231456&seqNum=2
"When AirDrop is enabled, it triggers Bluetooth to look for other AirDrop-enabled iOS devices that are in the area. This process is similar to pairing your iPhone with your car’s Bluetooth system. The devices “advertise” that they are open for business, and if they are within range of each other, then they start the connection process. Once the connection is made, a simple device-to-device Wi-Fi link, called Wi-Fi Direct, is created between the two. The Wi-Fi Direct link can be created even if you are in the middle of nowhere, without a Wi-Fi network. So, you get the fast file transfer speed of Wi-Fi without a router or an internet connection".
Enjoy! -------------------/Ivan