I think it probably evolved from a 1/2 wave horizontal dipole which has an approximate impedance of 75 ohms. 1/2 wave is chosen since it presents a reasonable real impedance at the feed point. Replace one half of the dipole with a ground plane and you have a 1/4 wave length antenna fed against a ground plane with an approximate impedance of 37 ohms.
Actually, a 5/8 wave length vertical is the best length because the major lobe of radiation is the lowest angle to the horizon, hence you get the longest distance. However, it needs a matching network to transform the feed impedance to something common like 50 ohms.