iMac not showing Airdrop discoverable options
I have same issue, Charlie. My iMac is a late 2015 27" version.
If you have the same version and problems, it's probably a OS issue. Hopefully users report enough and force Apple aware of the issue in coming OS update.

My solution (somehow worked):


Make sure you have a connected Internet Cable or Wifi 
Erase the Mac OS disk: press and hold CMD + R whilst the iMac start. Release since you see the Apple icon. Wait (it takes couple minutes)
Log-in screen will show several options then, including Disk Utility. Select it -> click on the Macintosh Disk in next windows (be careful: NOT CHOOSE THE HIDDEN IMAGE DISK) -> click Erase
Exit the Disk Utility windows
Now, you are in the initial screen. Choose ReInstall Mac. This takes you 3-4 hours with a good internet connection.
Your iMac after installation will be totally new. Keep in mind: Test AirDrop before installing ANY PROGRAM.


Unfortunately, as my observation, the AirDrop issue will happen again as soon as you begin to install certain softwares.

Hope Apple fix it soon.