Is the number of coin tosses of a probabilistic Turing machine a Blum complexity measure?
Probabilistic "functions" are not even functions in the classical sense of the term "function", that is, they do not assign to each member of the domain a specific member of the range. They are not even partial functions.  So the requirement of the Blum axioms that the complexity functions be computable cannot be satisfied.  

Specifically, for a TM $M$ with input $x$, the Blum axioms require an algorithm that for input $(M,x,n)$ decides if $\Phi(M,x)=n$, where $\Phi$ is the complexity measure.  But if $\Phi(M,x)$ is a probabilistic function, then it cannot be computed by an algorithm, and there also can be no algorithm as required by the axioms.

We could, however, define a complexity measure differently on probabilistic models, allowing a random variable, which is after all, a kind of function.  Then various properties of the r.v. -- mean, median, standard deviation, etc -- could well qualify as complexity measures, and you could develop the theory from there, in the probabilistic/statistical realm.

So, although Blum measures are somewhat out of favor, except for the classic space and time measures, there's no inherent reason the basic idea can't be extended in ways like this.