Which frame of reference is used for rest mass calculation? If there are no special reference frames or absolute rest frames, then rest mass of, say electron, is defined with respect to which frame?
Why cannot such a frame be used as the absolute rest frame?
 A: A better term instead of “rest mass” is “invariant mass”. This quantity is defined by the equation $m^2 c^2=E^2/c^2-p^2$ where m is the invariant mass, E is the total energy, p is the momentum, and c is the speed of light. 
This quantity can be calculated in any inertial reference frame and will give the same value. This is the quantity that particle physicists refer to when they talk about the mass of an electron or photon or other particle. It is the same as the rest mass, but the term invariant mass is preferred since it is more precise and general
A: If I understand your question correctly, I believe you are referring to an out-dated understanding where mass is a function of velocity.  If that was true, you would be correct that the definition of mass would require a velocity.
A better understanding is that mass does not change with velocity, mass is invariant.
Instead, the energy and momentum equations must be modified with a $\gamma$ term that accounts for the relativistic effects, such as 
$$E=\gamma m c^2$$
where
$$\gamma = \frac{1}{\sqrt{1-(v/c)^2}}$$
when $v=0, \gamma=1$
as $v \rightarrow c, \gamma \rightarrow \infty$
Defining mass as a function of velocity is an out-dated concept, but is unfortunatey prevalent in many low-level and high-school texts.  To convince yourself that it is wrong, consider the issue that velocity is a vector, so if mass was a function of velocity, then mass would also have to be a vector.
A good explanation is given in the review paper:
"The Concept of Mass in the Einstein Year", 
L. B. Okun (2008)
http://arxiv.org/pdf/hepph/0602037.pdf
A: For a specific particle there is a frame in which the particle is in rest. But there is no frame that is in rest in universe.
