It seems to be a game of broken telephone with Boltzmann's formula for the entropy at the beginning, and Fisher's Genetical Theory of Natural Selection (1930) in the middle. Fisher's statistical measures of fitness are the reproductive value denoted $v$, "the present value of the future offspring", by analogy to compound interest in economics, and what he calls the Malthusian parameter denoted $m$, "the relative rate of increase or decrease of a population when in the steady state". He then relates the latter to the gene frequencies $p,q$ (pp.34-35):
"The two groups of individuals bearing alternative genes,
and consequently the genes themselves, will necessarily either have
equal or unequal rates of increase, and the difference between the
appropriate values of $m$ will be represented by $a$, similarly the
average effect upon $m$ of the gene substitution will be represented
by $\alpha$. Since $m$ measures fitness to survive by the objective fact of
representation in future generations, the quantity $pqa\alpha$ will represent
the contribution of each factor to the genetic variance in fitness;
the total genetic variance in fitness being the sum of these contributions... whence it follows that, $\alpha dp =pqa\alpha dt$ and, taking all factors into consideration, the total increase in fitness,
$\Sigma (\alpha dp) = \Sigma (pqa\alpha)dt = Wdt.$
If therefore the time element $dt$ is positive, the total change of fitness
$Wdt$ is also positive, and indeed the rate of increase in fitness due
to all changes in gene ratio is exactly equal to the genetic variance of
fitness $W$ which the population exhibits. We may consequently state
the fundamental theorem of Natural Selection in the form
The rate of increase in fitness of any organism at any time is equal
to its genetic variance in fitness at that time."
So for Fisher $W$ is not the fitness itself, but the genetic variance of fitness. Why he chose $W$ Fisher does not say, but he later discusses the analogy to the second law of thermodynamics, as derived from the statistical mechanics. This, of course, reminds us of the Boltzmann formula for the entropy $S=k\ln W$. If $p(t)=p_0e^{mt}$, as in the Malthusian model, then $p(t+1)=e^{m}p(t)$, and with $W$ as defined today we have $m=\ln W$.
$W$ in the Boltzmann formula is already a case of the broken telephone. It was intended to stand for the Wahrscheinlichkeit (probability in German) of a macrostate for some distribution of microstates, but was misinterpreted early to stand for the number of microstates, as it does now. If this is what he had in mind, Fisher's use is more true to the original since $W$ is often described as the probability "that the individual will be included among the group selected as parents of the next generation" (Maynard Smith). Fisher writes (p.36):
"It will be noticed that the fundamental theorem proved above
bears some remarkable resemblances to the second law of thermodynamics.
Both are properties of populations, or aggregates, true
irrespective of the nature of the units which compose them ; both are
statistical laws ; each requires the constant increase of a measurable
quantity, in the one case the entropy of a physical system and in the
other the fitness, measured by $m$, of a biological population."
