Why is the voltage drop across this 1 ohm resistor being neglected in this practice problem? (Thevenin's Theorem)
You're trying to find the Thevenin equivalent voltage of the network. 

This is the voltage that the network produces when the output is open. 
When the output is open, no current flows out or in to the a terminal.
Therefore no current flows through the 1 ohm resistor.

Since  no current is flowing through the 1 ohm resistor, the voltage across it is 0, by Ohm's law.