Q: Airdrop

Have you tried turning Bluetooth off on both devices? I don't believe it's needed when using AirDrop.

Seems it requires both Bluetooth and Wi-Fi to be enabled.

Here's a troubleshooter.  Review the suggestions here:

http://support.apple.com/kb/HT5887

As the article cited above says:

"If you still can't locate a nearby user using Contacts Only, consider setting AirDrop to Everyone."

... and by the way, do both Minis have iOS 7?

It won't hurt one bit to try to reset BOTH devices.

Device Reset (won't affect settings/data/music/apps/etc)

1. Press and hold (& continue to hold) BOTH the Sleep/Wake button & the Home button.

2. Continue to hold BOTH (ignoring any other messages that may show) until you see the Apple logo on the screen.

3. Release BOTH buttons when you see the Apple logo and allow the device to boot normally.

Also check to see that both devices are able to connect to Wi-Fi (make sure the Wi-Fi radio is operating correcty)...any access point or otherwise and both are able to connect to Bluetooth devices (like a headset) without issue, eliminating those variables.