What does the Emergency Avionics switch do?
A quick google will turn up some answers. The consensus seems to be that it is just another switch in parallel with the Avionics Master Switch, such that, if the master switch fails, the "emergency" switch can be used to connect the juice instead! The point being that the master switch is a single  point of failure for the entire avionics suite (radios, transponder and nav) - if that switch fails in flight, you lose everything!

EDIT:

From the Piper Dakota (PA-28-236) Maintenance Manual:


  Electrical power for various avionics components is controlled by an avionics master switch located near
  the top of the instrument panel between radio stacks. It controls power to all radios through the aircraft master
  switch.
  
  An emergency bus switch is provided to supply auxiliary power to avionics bus in event of a radio master
  switch circuit failure. The emergency bus switch is located behind lower right shin guard, to left of the circuit
  breaker panel.