Can ZF prove choice over families of sets of ordinal definable sets?
Solution 1:
The class $\sf OD$ is just the closure of $\{V_\alpha\mid\alpha\in\rm Ord\}$ under Gödel operations (see Jech's Set Theory section on "Ordinal Definable Sets").
This means there is a definable well-ordering of $\sf OD$ by means of recursive construction of the definition.
Therefore, if $\bigcup X\subseteq\sf OD$, then there is a definable choice function from $X$.