How to remove mains hum from a BNC cable?
There are a few things you can do (any one of which may or may not work, and I would try them in this order - don't remove previous steps, just keep adding on):


Ground everything to a common ground
ferrite chokes
isolation transformers
invest in higher quality cables if shielding is inadequate
use a 55Hz 20Hz-wide notch filter to remove the hum (not preferable, but as a last resort)


Also, see this website: ARRL RFI.  I am a member of this organization and they know what they are talking about.