This may be a longshot, but I wonder if the problems of fair countably infinite lotteries and nonmeasurable sets aren't species of the same problem, for the following reason. The standard construction of nonmeasurable sets is the Vitali sets. But the Vitali sets on [0,1) have the following property: There is a countable collection of disjoint Vitali sets whose union is [0,1) and which are such that each of the sets is a translation (modulo 1) of every other set in the collection. Since we want translation-equivalent sets to have the same probability, this collection of Vitali sets implements a fair lottery with countably many tickets--the dart tossed at [0,1) intuitively has the same probability of hitting any particular Vitali set.
Here's another way to put it. Suppose we had a plausible probability for the individual outcomes of a fair countably infinite lottery. Then we could reasonably assign that value to the probability of an infinitely thin dart uniformly aimed at [0,1) hitting any given Vitali set.
(A Vitali set in this context contains exactly one element from each equivalence class of numbers in [0,1) under the equivalence relation x~y if and only if y is a translation of x, modulo 1, by a rational amount; the collection in question consists of any given Vitali set plus all its translations by rational amounts in [0,1).)