What's meaning of velocity of electron? How do we define velocity of microscopic particles at small scales. As for macroscopic particle we can scale the distance travelled by object. But this is not the case with electron.
Still, while explaining why mercury is liquid, people uses 'relativistic effect'.
 A: Special relativity (SR) was developed almost twenty yeas before the birth of quantum mechanics (QM). So, it was formulated in a classical mechanics language, assigning a key role to the concept of velocity. It is usual to estimate the importance of relativistic effects, on the basis of the ratio $v/c$ between the speed of the system and the speed of light.
In QM, the classical definition of velocity is untenable, since it is not compatible with the consequences of the theory on measurements of positions and momenta (uncertainty relations). However, one can introduce a "quantum velocity" analogous of the classical concept in term of momentum. For example, in absence of electromagnetic fields, by using the momentum operator divided by the mass. That is a quantity which becomes the velocity in the classical limit. Therefore, even if it is not meaningful to speak about the velocity of a particle, it is possible to evaluate a related quantity: the expectation value of $p/m$.
QM has been modified to take into account the constraints of SR. The resulting relativistic QM is not based on the concept of velocity, incompatible with the conceptual and formal structure of QM, but it is fully compatible with the SR relation between energy, mass and momentum:
$$
E^2=c^2p^2+m^2c^4.
$$
Thus, by using the SR relation 
$$
\frac{v}{c} = \frac{c p}{E}
$$
and using the well defined quantum expectation value  $\langle p\rangle$ in an eigenstate of the Hamiltonian, it is possible to provide an estimate of the importance of relativistic effects in a coherent QM conceptual frame, without using the definition of velocity of classical mechanics.
In practice, in electronic structure calculations, the signal of the importance of relativistic effects can be directly judged by comparing results of calculations by using relativistic and non-relativistic equations. If they differ, relativistic effects play a role.
A: The charge of the electron is measured in the Millikan oil drop experiment
The mass of the electron was measured in cathode ray tubes.
There are experiments where electrons can be seen and their momentum measured, in all particle physics experiments.
Here is a bubble chamber picture, where the magnetic field perpendicular to the plane of the picture allows measuring the momentum.


Bubble chamber photograph of an electron knocked out of a hydrogen atom
The curly line was produced by an electron that was struck by one of twelve passing $K^-$ beam particles in a liquid hydrogen bubble chamber. It curves in an applied magnetic field and loses energy rapidly, spiralling inwards.

The small curls are also knocked out electrons. The ionisation that allows to see the tracks are electrons just ejected from the hydrogen atom.
Once  the momentum is known the velocity is known.
For electrons still bound in a lattice , using  the quantum mechanical energy levels one can give an average velocity for the electrons in a specific orbital, but this is not particularly useful.
It is not clear what you mean by:

Still , while explaining why mercury is liquid, people uses 'relativistic effect'

You must mean this

Relativity states that objects get heavier the faster they move. In atoms, the velocity of the innermost electrons is related to the nuclear charge. The larger the nucleus gets the greater the electrostatic attraction and the faster the electrons have to move to avoid falling into it. So, as you go down the periodic table these 1s electrons get faster and faster, and therefore heavier, causing the radius of the atom to shrink. This stabilises some orbitals, which also have a relativistic nature of their own, while destabilising others. This interplay means that for heavy elements like mercury and gold, the outer electrons are stabilised. In mercury’s case, instead of forming bonds between neighbouring mercury atoms, the electrons stay associated with their own nuclei, and weaker interatomic forces such as van der Waals bonds hold the atoms together.

This is the average velocity of electrons bound in orbitals, as mentioned above. Orbitals are probability loci for the electrons in specific energy levels. As the energy levels are known or calculated an average velocity for the electron can be derived. The link refers to a calculation   with a quantum mechanical model that fits the behavior of mercury :

They showed that if they ignored relativity when making their calculations, the predicted melting point of mercury was 82°C. But if they included relativistic effects their answer closely matched the experimental value of -39°C.

