Devices not showing up for AirDrop
Steps 3 and 4 are the trick!

Similar issue:


Can send file from MBP to iPad.
Can not see MBP on iPad* airdrop screen.


Fix that worked for me:


On iPad: turn on Wifi, Bluetooth and AirDrop (share with Everyone)
On Mac: turn on Wifi, Bluetooth and AirDrop (share with Everyone), then try to share the file(s) on AirDrop
On Mac: Bluetooth > Browse files on Device; select Jeff's iPad, then 'Browse'; notice it connects, but then it drops right after with the message "The device does not have the necessary services" and 
On iPad: observe that 'Mike's MacBook Pro' shows on AirDrop
MBP', despite of the error (another error message might be shown under the MBP's name)
On iPad: send pictures/videos to Mac.
[extra] On iPad and MBP: for security reasons, turn off AirDrop (and maybe Bluetooth too) once you're done!

*Obs: it should work on iPhone too;  I upvoted this answer since it helped me to find this solution .