How much energy is needed to create an electron? I know how to calculate the electrostatic energy of a sphere (it has a well defined radius). But how can I calculate the electrostatic energy of an electron as it is a point particle? By electrostatic energy of a charge distribution we generally mean the energy needed to place the charges where they are. (In short energy needed to create the system). 
So, how much energy I need to create an electron?
 A: I want to complete the other answer by addressing the difference of a charge distribution made statistically up by a huge number of electrons , and what an electron means:
At the level of elementary particles, one of which is the electron, there  are no charge distributions, as elementary particles are point particles, and charge is a quantized quantity that is an attribute of the electron, like spin and its mass that cannot be further  cut up. In the case of a proton, there exists a charge distribution given by the distributions of the quarks that compose it: the quarks are the elementary particles in this case.
A lone electron cannot be created because of lepton number conservation, it has to come with an antielectron or be part of a decay process, like the decay of the muon where the electron number is conserved by the creation of an antineutrino-electron.
A: The rest mass energy of an electron is 511 keV. So this is how much energy is required to create an electron, at rest, and which is not in an electromagnetic field. You need not be concerned about the point-like nature of the electron, that just makes it simpler to calculate the difference in ths energy (if necessary) caused by any external electric potential $\phi$ where the electron is. The electrostatic energy is just $-e \phi$.
In practice it is a bit more complicated depending on how you intend to create your electron because of conservation laws. i.e. charge, lepton number and momentum must be conserved, as well as energy, so the negative charge must have been taken from somewhere else or a positively charged particle must also have been created.
In the case of pair production, a photon can produce an electron, but it needs to interact with a nucleus to conserve momentum and must also produce a positron to conserve charge and lepton number. In this case the minimum energy required for the photon is the rest mass energy of the electron plus the same rest mass energy for the positron, (i.e. 1022 keV), and even more if it turns out that these particles have kinetic energy.
A: If you try to concentrate separate electric charges repelling each other with the Coulomb forces, you would never succeed because that would require infinite amount of energy. That means if electron is charge concentrated into a point, it is not made of smaller parts but is an elementary particle and its rest energy is not electrostatic. This is the common viewpoint. (It is very unlikely but theoretically at very small distances the repulsive force could be weaker than what the Coulomb law says so finite energy would be sufficient to squash the charges into small region or a point. Part of rest energy would be electrostatic then.)
