Let $X$ be a topological space such that there exists a collection of meager clopen sets $(C_i)_{i \in I}$ such that $X = \bigcup_{i \in I} C_i$. I want to prove that $X$ is then meager itself.
As each $C_i$ is meager there is a sequence of nowhere dense sets $(N_{i,n})^\infty_{n=1}$ such that $C_i = \bigcup^\infty_{n=1} N_{i,n}$. For natural $n$ define set $M_n = \bigcup_{i\in I} N_{i,n}$. Then by basic set theory
$$ X = \bigcup_{i \in I} C_i = \bigcup_{i \in I} \bigcup_{n=1}^\infty N_{i,n} = \bigcup_{n=1}^\infty \bigcup_{i \in I} N_{i,n} = \bigcup_{n=1}^\infty M_n. $$
So, we need to show that each $M_n$ is meager and we are done. Initaly I thought that it may be possible to use clopennes to write something like
$$ \mathrm{int}\;\mathrm{cl}\;M_n = \mathrm{int}\;\mathrm{cl}\;\bigcup_{i \in I} N_{i,n} = \bigcup_{i \in I} \mathrm{int}\;\mathrm{cl}\; N_{i,n} = \bigcup_{i \in I} \emptyset =\emptyset, $$ where $\mathrm{int}$ and $\mathrm{cl}$ stand for interior and closure operations respectively, so $M_n$ is nowhere dense. But this works only if $(C_i)_{i \in I}$ are disjoint. This won't work in general, for example take $(C_i)_{i \in I}$ to be a collection of all clopen sets in $X=\mathbb{Q}$, and choose $N_{i,n}$ to be a collection of singletons so each $M_n = \mathbb{Q}$. I know that in this example sets $X$ is meager, but I wanted to show that argument above may fail in general.
My first idea was to transform $(C_i)_{i\in I}$ into a different disjoint clopen cover $(C'_j)_{j \in J}$, such that for each $j\in J$ there is $i\in I$ with $C'_j\subset C_i$. But how to do so? In case $X$ is locally connected one can just take its connected components, but this won't work in general. I thought that a transfinite induction may also work. By abuse of notation assume that $I$ is an ordinal, let $\mathcal{C}_{1}$ be an well-order by $I$ version of $(C_i)_{i\in I}$. then for a non-limit ordinal $\kappa + 1$ we may construct a Set $\mathcal{C}_{\kappa + 1} = \{ C^\kappa_{1}, \ldots, C^\kappa_\kappa \} \cup \{ C^\kappa_{\kappa +1} \setminus C^\kappa_\kappa , \ldots, C^\kappa_{I} \setminus C^\kappa_\kappa \}$, which is an ordered set of clopen sets, where each set in the first brackets is disjoint from any other set in $\mathcal{C}_{\kappa + 1}$. And for the limit ordinals $\lambda$ just write $\mathcal{C}_\lambda = \{ C^i_i | i < \lambda\} \cup \left\{ \bigcap_{i < \lambda} C^i_j |\lambda \le j \le I \right\}$? I don't think this will work, as clopeness may be lost. But in the end I hoped $\mathcal{C}_I$ to be a disjoint cover I seek.
Please, help me to prove this statement.
Here's the trick. Let $(U_j)_{j\in J}$ be a maximal family of disjoint meager open subsets of $X$ (such a maximal family clearly exists by Zorn's lemma, for instance). Let $U=\bigcup U_j$. Then $U$ is meager using the line of argument you propose, since the $U_j$ are disjoint. But now observe that $X\setminus U$ is closed and has empty interior (if a point $x$ was in its interior, then we could find a meager open neighborhood of $x$ contained in it which we could then add to our family, contradicting maximality). So we can add $X\setminus U$ as one more nowhere dense set to witness that $X$ is meager.
(Incidentally, this argument only requires the $C_i$ to be open, not necessarily clopen. So this shows that if a topological space is locally meager (i.e., each point has a meager neighborhood) then it is meager.)