In symbols, A is the symbol for affinity, chemical affinity, or elective affinity. The symbol "A", in thermodynamic formulations, can also refer to mechanical work from the German “Arbeit”, meaning work, and to Helmholtz free energy, a thermodynamic formulation of affinity.

Affinity
It is difficult to say when exactly A can into use as “affinity”, being that this subject, according to English chemistry historian James Partington,

The start of the science of modern affinity chemistry, however, traces to descriptions of chemical "attractions" by English physicist Isaac Newton in "Query 31" to his 1718 Optics. [1] In the century to follow the terms affinity and attraction were often used synonymously.

In 1936, Belgian physicist Theophile de Donder had clearly used the symbol "A" for affinity as the negative partial of the partial of the Gibbs free energy per unit partial of extent of reaction for a change in a isothermal isobaric system: [2]

$A=-\left(\frac{\partial G}{\partial \xi}\right)_{p,T}$

In modern simplified notation:

$~ A = - \Delta G ~$

In addition, the standard affinity, symbol A°, is the affinity of a chemical reaction when each constituent of the reaction is take in its standard state. [3]

Arbeit
In 1864, French chemist Marcellin Berthelot put forward his theory that the heat of reaction was the true measure of affinity; and later, in circa 1775, proposed his maximum work principle, which argued that that the "maximum work" that a chemically reactive system can produce was a measure of the heat released from the chemical reactants on going to products. This view, however, soon came into conflict with the new 1865 "entropy" state formulation of heat by German physicist Rudolf Clausius.

In 1882, to reconcile the inconsistencies between Berthelot's view and Clausius' view, Hermann Helmholtz, in his "On the Thermodynamics of Chemical Processes", proved that the true measure of affinity is free energy (not heat), thus showing Berthelot's theory to be fallacious (at all temperatures above absolute zero). In the period to follow this date would be a probable time for symbol introduction, as various free energy formulas began to be used. Since the German word for work is “Arbeit”, likely used by Helmholtz, the symbol “A” soon began to be used for work, maximal work, free energy or Helmholtz free energy, and also affinity.

In 1917, Walther Nernst was using A for “work”.
In 1924, James Partington was using A for “work”.

Reaction energy
In 1905, Fritz Haber was using A for “reaction energy”.

Helmholtz free energy
In 1923, Gilbert Lewis, building in some sense on Haber, was using A for “Helmholtz free energy”, as follows:

A = U – TS

In circa 1950s, IUPAC began to assign A to Helmholtz energy (Helmholtz free energy), a function which measures affinity for isothermal isochoric systems.

At some point along the line, however, this A for Helmholtz free energy symbol notation, as shown in the characteristic function notation table, began to be supplanted by the symbol F for Helmholtz free energy (or "free energy", for short, with an V,P constant system assumed).

References
1. Muir, Matthew M.P. (1907). A History of Chemical Theories and Laws (ch. XIV: Chemical Affinity, pgs. 379-430, esp. keyword: “Bergmann”, pgs. 384-94). Wiley.
2. De Donder, Theophile. (1936). Thermodynamic Theory of Affinity: A Book of Principles (pg. 2). Oxford University Press.
3. Perrot, Pierre. (1998). A to Z of Thermodynamics (Standard affinity, pg. 284). Oxford University Press.
4. Partington, James. (1937). A Short History of Chemistry. Dover.