Actually, Avogadro's number was not selected per se, it was calculated after experimental data on the charge of an electron was gathered.
An accurate calculation of Avogadro's number became possible after American physicist Robert Millikan successfully measured the charge on a single electron in his famous oil drop experiment.
This value was used with the charge on a mole of electrons, which had been known at that point, to calculate Avogadro's number. The charge on a mole of electrons is known as the Faraday constant, and equal to
The charge on a single electron was determined to be
Avogadro's number was calculated by dividing the charge on a mole of electrons by the charge of a single electron: