Here is an answer for Gerhard's (easier) special case, where the open sets are disjoint. 

<b>Claim.</b> The union of an arbitrary family of pairwise disjoint open meager sets is meager. 

Proof. Suppose that $U_i$ are pairwise disjoint and meager, so that $U_i\subset\bigcup_n C_n^i$, where each $C_n^i$ is closed and nowhere dense. Let $C_n=\bigcup_i (C_n^i\cap U_i)$. This set is not dense on any open set, because if it were dense on some nonempty $V$, then it would be dense on some nonempty $V\cap U_i$, but that is impossible since by the disjointness hypothesis only $C_n^i$ contributes points to this set, and it is nowhere dense. Thus, the closure of $C_n$ is closed and nowhere dense, and $\bigcup_i U_i$ is contained within $\bigcup_n C_n$, since each $U_i$ is contained within and is in fact equal to $\bigcup_n (C_n^i\cap U_i)$. So $\bigcup_i U_i$ is meager. QED

Perhaps a proof for the general case can be made by putting the sets $C_n^i$ together more carefully.