As it has been already pointed out, ''logistic'' comes from logistic curve/function/distribution (which is underlying logistic regression). So the question is: where is logistic coming in their names?
The reference to Verhulst (i.e. Wikipedia's statement) seems a bit false. While it is clearly true that it is most widely attributed to Verhulst, the first actual use seems to come from Edward Wright. See Thompson: On Growth and Form (1945), page 145. (Found via the well-known Earliest Known Uses of Some of the Words of Mathematics page.)
Thompson hints that Verhulst used it in connection with its S-shape, but gives no clue about Wright.
However, given that one of the most important parts of Wright's work was pertaining to logarithms, it seems logical that he used it as a reference to logarithm. Indeed (and more precisely), the 1911 edition of Encyclopaedia Britannica refers to the old mathematical term logistic number:
The old name for what are now called ratios or fractions are logistic numbers, so that a table of log (a/x) where x is the argument and a a constant is called a table of logistic or proportional logarithms; and since log (a/x) =log a-log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign.
Also note that logarithm itself comes from proportion (logos) + number (arithmos); originally coined by John Napier.
So, I believe, this is the most likely explanation: ''logistic'' is used in Wright's time in connection with what we now call ''logarithm'', which was used by Wright when he constructed that curve.