There are two kinds of momentum for the electron, intrinsic spin momentum, which it always has, and angular momentum, which what you are taking about. An electron does and always will have, mass, spin angular momentum, and electric charge.
If you think of the electron as something like a marble, you will get the wrong picture of both its spin angular momentum and its orbital angular momentum.
Don't think of the electron going around the nucleus, like a planet going around the Sun. At the quantum level, its all about forgetting classical concepts and filling your brain insead with mathematical ones, because common sense no longer applies.
The only thing that is of any importance is the electron's wavefunction. The electron has orbital angular momentum if the wavefunction, which describes it's position, allows for it. That is the basis for whether or not, the electron has orbital angular momentum. Some electrons do, some don't. In the hydrogen atom , the ground state electron has no orbital momentum. If you try to think classically, this makes no sense. If you read the Wikipedia article linked below, it might.
The orbital angular momentum has no relation to mass, that's because the electron has mass and it's embedded in us to think, if it has mass and spins around, it's got angular momentum. No, not in the quantum world.
The orbital angular momentum itself has strong fluctuations and it is zero only on average. If you find it impossible to stop think of it in terms of planets, then think that the electron is orbiting around the nucleus, but this orbiting does not have a prefered trajectory, i.e. it goes in all directions around the nucleus.
I was going to include a picture of orbitals of electrons, but that would just bring the classical picture into your mind again.
In his answer, dmckee says:
It really means that the state of the system is an eigenstate of the total angular momentum operator that corresponds to a measurement of 0.
This is an example of dropping the classical world picture, and instead applying mathematical language to describe the quantum world.
Have a read of this Angular Momentum Wikipedia., but please DON'T take the illustrations literally.