How is the degenerate electron gas state "degenerate"? What is "degenerate" in the degenerate electron gas state?   
Why is it called degenerate?
 A: A degenerate gas is one where more than one electron (in fact two, one in each spin state) occupies each possible low-energy state up to the Fermi energy. I suppose the term "degenerate" comes from the multiple occupancy of each energy level.
A: A normal gas consists of particles that do not interact much except for elastic collisions.
Often, describing a gas in a simplified way ignoring the other interactions completely is good enough.
This simplified model is the "ideal gas".
This simplified description fits for the electron gas also, as long as the pauli exclusion principle does not become relevant.
This would be called an ideal Fermi gas.
But there are cases where the pauli principle is very relevant, when not enough low energy levels are free for electrons.  
When electrons are confined into a limited volume, the pauli principle limits the number of electrons that can occupy that space for a given maximal energy level.  
Now, the electron gas is degenerate in that there is another interaction apart from what normally makes a gas - and this can be the dominating property of the degenerate gas.
For example, the pressure of a degenerate electron gas is higher than one would expect from a gas.
The degenerate electron gas does not only behave differently from an ideal gas, it has fundamentally different properties based on quantum effects.

See Richard Fitzpatrick "Quantum Mechanics"
Three-Dimensional Quantum Mechanics/Degenerate Electron Gases
