Given a metric space $(X,d)$,a probability measure $\mu$ (on the Borel sigma algebra) and an open cover $C:=\{A_i\}_{i\in I}$ of $X$, is it always possible to find a countable subset of $C$ that covers $X$ up to a null set i.e. does there exists countable $C'\subseteq C$ such that $\mu(X\backslash \cup_{A\in C'} A)=0$?
Naively one would suspect that this is true because if we consider a "minimum" $C'\subseteq C$ that covers $X$ up to a null set, then for any $A\in C'$ we have $\mu(A\backslash \cup_{A\neq A'\in C'} A')>0$. Hence if such $C'$ is uncountable it's routine to show that $\mu(X)=\infty$, a contradiction. Obviously this argument doesn't necessarily work because such $C'$ may not exists.
For the sake of completeness I should mention that this is a natural question that one might want to answer to solve Exercise 2.2.3 of Einsiedler and Ward's Ergodic Theory text: Let $(X,d)$ be a metric space, $T:X\to X$ Borel measurable, and $\mu$ be a $T-$invariant probability measure. Prove that for $\mu-$almost every $x\in X$ there is a sequence $n_k\to\infty$ with $T^{n_k}(x)\to x$ as $k\to \infty$
I don't have a complete answer, but I will argue that a positive answer to your question would refute a statement "widely believed by set theorists", namely the existence of measurable cardinals.
Let $X$ be an uncountable set with metric \begin{align} d(x,y) &= \begin{cases} 1 & \text{if $x\neq y$,} \\ 0 & \text{if $x=y$,} \end{cases} \end{align} so that $X$ has the discrete topology.
Suppose we can find a probability measure $\mu$ such that $\mu(\{x\})=0$ for each $x\in X$. Then, this would provide a negative answer to your question, namely, the open cover $\{B_{1/2}(x): x\in X\}$ would have no countable subfamily that covers $X$ up to a null set.
According to this answer and the Wikipedia page it cites, the existence of such a measure space cannot be proven from ZFC (assuming the consistency of ZFC). If you can prove in ZFC that countable subcovers modulo null sets always exist, you would get a proof of the inconsistency of the existence of measurable cardinals with ZFC. This is a possibility as far as I understand, but it seems that most set theorists do not consider it a likely possibility (see e.g. here or Volume 5II, Chapter 54 or Measure Theory by Fremlin).