Is a low hum coming from a new, quality ceiling fan caused by an inappropriate wire gauge? The house wiring is only 9 years old and the fan outlet was specifically wired for a fan.
Is it the 60 cycle AC causing the audible hum? If so, how do I mitigate it? The sound didn't occur when it was first installed several months ago but has just started.

I think its your fan. To verfiy that if its on a switch check the switch with a volt meter then check at the fan for correct voltage. Is the fan balanced and not putting side tourqe on the motor. You can buy weights for this or use popcicle sticks on the top of the blades to blance. How old is the fan? Does the hum changes at all speeds?

And what kind of controls are you using to change speeds? The chain? A slide switch on the wall (looks kind of like a dimmer)? A remote? Something else?

It's not the wiring.

I'll echo Zoe's question. What controls are you using? Check also for loose screws and parts.

See:

General ceiling fan troubleshooting and help FAQ- Ceiling Fans N More