I think there is no room for fidgeting but then they are still subjected to uncertainty principle, my question is when the electrons are sardine-packed to the extreme so that Pauli exclusion principle set in preventing them from occupying the same quantum state (lowest), do they gain any mass?
The electron is an elementary particle, one of its intrinsic properties is its rest mass. During electron degeneracy pressure, the rest mass of the electrons does not change.
Though, their momentum does change. They do gain kinetic energy (momentum) during electron degeneracy pressure. This is due to the HUP, which states, that a material subjected to ever increasing pressure, will compact more, the electrons within will become more localized, thus, they will gain momentum.
Electron degeneracy pressure is a particular manifestation of the more general phenomenon of quantum degeneracy pressure. The Pauli exclusion principle disallows two identical half-integer spin particles (electrons and all other fermions) from simultaneously occupying the same quantum state. The result is an emergent pressure against compression of matter into smaller volumes of space. Electron degeneracy pressure results from the same underlying mechanism that defines the electron orbital structure of elemental matter.
You might be thinking about relativistic mass, but it is not used really, because it is sometimes confusing.
There is no controversy or ambiguity. It is possible to define mass in two different ways, but: (1) the choice of definition doesn't change anything about predictions of the results of experiment, and (2) the definition has been standardized for about 50 years. All relativists today use invariant mass.