Does upgrading Wireless Card make AirDrop work?

Simply upgrading the wireless card in a device, if possible, makes the feature available. (If you're minded to do this, try to ensure that you choose a replacement that has the same number of antenna connectors as the card that you are replacing.)

In order to use AirDrop, the wireless chip must have the ability to connect to an infrastructure network and be part of a personal area ad-hoc network at the same time.

This is known to be supported out of the box via:

Atheros AR5BHB92

Atheros AR5BXB92

Atheros AR5B93

Atheros AR5BXB112

Broadcom BCM94322HM8L

Broadcom BCM94322MC

With a modified kext, it is possible via:

Atheros AR5B95

Atheros AR5BXB72

Cards that are known to not support AirDrop:

Broadcom BCM94321MC

Broadcom BCM94312MCG

Broadcom BCM94312MCAG

Broadcom BCM94311MCG