Does the mass of an electron change with its "energy state"? When an electron absorbs a photon, it gets into a higher energy state and goes into the upper orbit/shell.
Does (rather should) this absorption of energy also have an impact on its mass (although incredibly small)?
Can we even measure the mass of an electron while it is it still bound to the nucleus?
 A: A fundamental particle's rest mass never changes. It's mass is a natural constant, and one of the numbers which uniquely identifies it (like its spin). On the other hand, the invariant mass of the atomic system does increase as the electron becomes excited, bringing the atom into a higher energy state. In that sense, the atom (not the electron) gets "heavier" because of the increased energy of the internal configuration of particles.
A: This is really an extended comment to Geoffrey's answer, so please upvote Geoffrey's answer rather than this.
The mass of a hydrogen atom is $1.67353270 \times 10^{-27}$ kg. If you add the masses of a proton and electron together then they come to $1.67353272 \times 10^{-27}$ kg. The difference is about 13.6eV, which is the ionisation energy of hydrogen (though note that the experimental error in the masses isn't much less than the difference so this is only approximate).
This shouldn't surprise you because you have to add energy (in the form of a 13.6eV photon) to dissociate a hydrogen atom into a free proton and electron, and this increases the mass in accordance with Einstein's famous equation $E = mc^2$. So this is a direct example of the sort of mass increase you describe.
However you can't say this is an increase of mass of the electron or the proton. It's an increase in mass of the combined system. The invariant masses of the electron and proton are constants and not affected by whether they're in atoms or roaming freely. The change in mass is coming from a change in the binding energy of the system.
A: In the example you site, you are talking about a bound electron. In this case, the electron does not gain (any kind of) mass because the energy of the photon goes into changing the state of the electron (to a higher energy state). This energy is "given back" when the electron returns to its previous state, giving off an equivalent photon. 
A: No. The mass of an electron does not change with its increased energy (by absorbing photon energy).
A photon does not have rest mass but energy, and such photon energy does not convert into mass, in whatever form - as per the equation E=mc^2.
The photon energy rather converts into higher frequency, not mass, as can be seen in Max Planck's equation E=hv (eich nu) or, E=hf. Photon energy rather converts into frequency, as a result such energy excited election should increase its energy and frequency of its spin around the nucleous, not increasing mass of the electron or mass of the atom itself, but increament in it's energy frequency.
A: Even a free electron gets heavier under the influence of accelerating photons. Best example are colliders where some amount of photons energy stay on the electron and some amount the electron looses again. Then faster the electron then higher the loose. Physics lives from models and interpretations and your interpretation is nice. It brings closer together the terms energy and mass. 
