Question

Do electrons have mass?

Answer 1

Michael Faraday was the first one to show that if electricity is passed through a solution of an electrolyte, chemical reactions occur at the electrodes. This suggested the particulate nature of electricity which meant electricity happens due to passage of few discrete particles passing closely and together.

Complete answer:
Before the discovery of subatomic particles, people believed that atoms are the smallest unit of matter but after performing his cathode ray experiment, particulate nature of electricity was discovered and in the cathode ray tube there were rays going from the anode side to the cathode side. These rays were negatively charged and therefore they were named electrons.
What we know after this is that the atom has to be neutral because the matter is never charged then there must be something to balance the charge. Protons were discovered by Chadwick, when he experimented with modified cathode ray tubes which produced canal rays which went from the cathode side to the anode side and therefore there were some positively charged particles.
Many atomic models came after that to propose what the charges were on one proton and one electron, which came to be the same according to the Millikan’s oil drop experiment.
After Millikan’s oil drop experiment, we have a ratio between the charge of electrons and their mass.
Because protons and neutrons have a mass of 1 amu each as known earlier, the mass number equals the sum of the numbers of protons and neutrons. As the mass of the electron was found to be very small, it was also suggested that the entire mass of the atom must comprise of positively charged particles.
Electrons have a mass of approx. $\dfrac{1}{{1836}}amu$and has a negative charge.
Yes, electrons have a mass.

Note:
Protons and neutrons are present in the nucleus of an atom and the number of protons is always equal to the number of electrons.