Rota used a lot of hyperbole, and "no one likes to bring up" is exaggerated in reference to some of those twelve problems. While Rota was writing this in 1998, he and I exchanged probably a couple of hundred emails about it. I convinced him to use the term "atomless" instead of "non-atomic" since the latter terms seems as if it ought to mean "not atomic", but that's not what it means. I also suggested that rather than merely stating the definitions, he should be completely explicit about the fact that the mapping goes in the opposite direction from what is conventional.
Normally a real random variable is defined as a measurable function $X:\Omega\to\mathbb{R}$, where $\Omega$ is the underlying set of a probability space. For every Borel-measurable subset $S$ of $\mathbb{R}$, you have a measurable set $\{\omega\in\Omega : X(\omega)\in S\}$. This mapping takes subsets of $\mathbb{R}$ to subsets of $\Omega$, rather than points of $\Omega$ to points of $\mathbb{R}$. That's the "opposite direction" aspect of it. But in pointless probability, you never consider points of $\Omega$, or consider $\Omega$. Rather, you have a Boolean algebra of propositions to which you want to assign probabilities. Via "Stone's duality", you can identify each such proposition with a clopen subset of the Stone space of the Boolean algebra.
The Stone space of a Boolean algebra $A$ is the space of Boolean homomorphisms from $A$ into $\{0,1\}$ with the topology of pointwise convergence of nets of such homomorphisms. Every such space is a totally disconnected compact Hausdorff space. Every totally disconnected compact Hausdorff space arises is the Stone space of the Boolean algebra of all of its clopen subsets.
In effect, instead of measurable subsets of $\Omega$, you've got clopen subsets of a Stone space, and you never assign a probability to an individual point in that space except when the set containing just that one point is clopen. Hence this is "pointless".
I don't actually know the original source of "pointless probability theory" (so this doesn't answer that question). The stuff on Boolean algebras and Stone spaces can be found in Halmos' Lectures on Boolean Algebras, which I think is in print again. Halmos never uses the term "Stone space"; rather he defines a "Boolean space" as a totally disconnected compact Hausdorff space. Halmos proves all the basic facts of "Stone's duality" between the category of Boolean algebras and Boolean homomorphisms, and that of Boolean spaces and continuous functions. The two categories are essentially opposites of each other, so there's the reversal of directions again.
Here's the finitary version of the duality: Suppose you have a mapping $f:A\to B$ where $A$, $B$ are finite sets. This is the continuous function from one Boolean space to another. There's a corresponding mapping $g:\mathcal{P}(B)\to\mathcal{P}(A)$ (except that in the general case instead of all subsets of $B$ and of $A$, you have all clopen subsets and you only consider continuous $f$) defined by $g(B_1) = \{x\in A : f(x) \in B_1\}$. $g$ will always be a Boolean homomorphism. In the finite case, that just means it preserves unions, intersections, and complements. In the general case, instead of union, you have the interior of the closure of the union---the smallest clopen set that includes the sets whose "union" is taken. You can show that either of $f$, $g$ is one-to-one if and only if the other is onto.
Rota said he intended to expand his "Twelve Problems" paper to about twice the length it had when he presented it as his "Fubini Lectures" in Torino. But he died suddenly in April 1999.