Actually it is ~ 1/137 that is referred to as "alpha" because "the fine structure constant was first identified and named in 1916 by Arnold Sommerfeld, Heisenberg's teacher, who was studying fine patterns in spectral lines. Alpha is central to Quantum Electrodynamics." (1).
For the history one can start with "The fine-structure constant before quantum mechanics," by Helge Kragh ...
"This paper focuses on the early history of the fine-structure constant, largely the period until 1925. Contrary to what is generally assumed, speculations concerning the interdependence of the elementary electric charge and Planck's constant predated Arnold Sommerfeld's 1916 discussion of the dimensionless constant. This paper pays particular attention to a little known work from 1914 in which G N Lewis and E Q Adams derived what is effectively a numerical expression for the fine-structure constant." (2).
also ... more history by Kragh (3).
The fine structure constant determines the strength of the electromagnetic interaction and is also related to the other forces of nature in fundamental ways not well understood yet. For more recent historical research see Arthur Miller's book Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung, in paperback 137: Jung, Pauli, and the Pursuit of a Scientific Obsession (4).
For new research on the meaning of the fine structure constant and more in-depth references see the paper by Michael Sherbon on "Mathematical Constants of Natural Philosophy" (5).
Are your ears still burning? How was dinner? ;-)