In the 19th century chemists found it puzzling that benzene could be so unreactive toward addition reactions, given its presumed high degree of unsaturation.  The cyclohexatriene structure for benzene was first proposed by August Kekulé in 1865. Over the next few decades, most chemists readily accepted this structure, since it accounted for most of the known isomeric relationships of aromatic chemistry.
Between 1897 and 1906, J. J. Thomson, the discoverer of the electron, proposed three equivalent electrons between each carbon atom in benzene.
An explanation for the exceptional stability of benzene is conventionally attributed to Sir Robert Robinson, who was apparently the first (in 1925) to coin the term aromatic sextet as a group of six electrons that resists disruption.
In fact, this concept can be traced further back, via Ernest Crocker in 1922, to  Henry Edward Armstrong,  who in 1890 wrote "the (six) centric affinities act within a cycle...benzene may be represented by a double ring (sic) ... and when an additive compound is formed, the inner cycle of affinity suffers disruption, the contiguous carbon-atoms to which nothing has been attached of necessity acquire the ethylenic condition".
Here, Armstrong is describing at least four modern concepts. First, his "affinity" is better known nowadays as the electron, which was to be discovered only seven years later by J. J. Thomson.  Second, he is describing electrophilic aromatic substitution,  proceeding (third) through a  Wheland intermediate, in which  (fourth) the conjugation of the ring is broken. He introduced the symbol C centered on the ring as a shorthand for the inner cycle, thus anticipating Erich Clars notation. It is argued that he also anticipated the nature of wave mechanics, since he recognized that his affinities had direction, not merely being point particles, and collectively having a distribution that could be altered by introducing substituents onto the benzene ring (much as the distribution of the electric charge in a body is altered by bringing it near to another body').
The quantum mechanical origins of this stability, or aromaticity,  were first modelled by Hückel in 1931. He was the first to separate the bonding electrons into sigma and pi electrons.