What is "logistic" about the logistic distribution, in a common sense way? What is the etymology of and the lexical rationale for the name, not just pure math definition?
The source document for the name "logistic" seems to be this 1844 presentation by P.-F. Verhulst, "Recherches mathématiques sur la loi d'accroissement de la population," in NOUVEAUX MÉMOIRES DE L'ACADÉMIE ROYALE DES SCIENCES ET BELLES-LETTRES DE BRUXELLES, vol. 18, p 1.
He differentiated what we would now call exponential growth of population when resources are essentially unlimited (as seen for example in the growth of the US population in the late 18th and early 19th centuries) from the slower growth when resource limits begin to be reached.
What we call exponential growth, however, he called a "logarithmique" curve (page 6).
He then developed a formula for population growth in the presence of resource limits, and said of the resulting curve:
"Nous donnerons le nom de logistique à la courbe..." which I translate as "We call the curve logistic..." (original emphasis).
That would seem to be intended to distinguish this growth pattern from the "logarithmique" growth in the absence of resource limits, as the figure at the end of the paper illustrates.
The specific form of the equation presented by Verhulst allows for an arbitrary upper asymptote (eq. 5, page 9), while the form we know and love in statistics is the specific case with an asymptote of 1.
(Cross-posted from History of Science and Mathematics: source of “logistic growth”?)
As Ed states, the term logistic is due to the Belgian mathematician Pierre François Verhulst, who invented the logistic growth model, and named it logistic (French: logistique) in his 1845 "Recherches mathématiques sur la loi d'accroissement de la population", p. 8:
Nous donnerons le nom de logistique à la courbe
We will give the name logistic to the curve
He does not explain why he uses this term, but it is presumably by analogy with arithmetic, geometric, and in contrast to logarithmic (per text and illustration that Ed includes).
The French term logistique is from Ancient Greek λογιστικός (logistikós, “practiced in arithmetic; rational”), from λογίζομαι (logízomai, “I reason, I calculate”), from λόγος (lógos, “reason, computation”), whence English logos, logic, logarithm, etc. In Ancient Greek mathematics, logistikós was a division of mathematics: practical computation and accounting, in contrast to ἀριθμητική (arithmētikḗ), the theoretical or philosophical study of numbers. Confusingly, today we call practical computation arithmetic, and don't use logistic to refer to computation.
Verhulst first discusses the arithmetic growth and geometric growth models, referring to arithmetic progression and geometric progression, and calling the geometric growth curve a logarithmic curve (confusingly, the modern term is instead exponential curve, which is the inverse), then follows with his new model of "logistic" growth, which is presumably named by analogy, after a traditional division of mathematics, and in contrast to the logarithmic curve. The term logarithm is itself derived as log-arithm, from Ancient Greek λόγος (lógos) and ἀριθμός (arithmós), the sources respectively of logistic and arithmetic.
There is no connection with logis (lodging), though that is the source of the term logistics (1830).
The logistic distribution is not a common distribution in analysis, but it ties together the notion of a latent underlying continuous variable which is thresholded in binary outcomes. It turns out that thresholding a logistic RV (to 1 if the RV is greater than some unknown value and 0 otherwise) and calculating a maximum likelihood leads to logistic regression. Contrast this approach with thresholding a normally distributed random variable which leads to probit regression. Applying multiple thresholds leads to cumulative link models.
Now, if your question concerned logistic regression, the term was coined by David Cox in 1958 "The regression analysis of binary sequences (with discussion)" in JRRS. He used the term to the logistic, sigmoidal shape of the modeled mean. For describing the process of a curve which models probabilities that accumulate according to a probabilistically sound way, the term "logistic" is an intuitive choice and the nomenclature stuck.