I'm fairly certain this question has a very simple answer, and that I've learned it before; I just can't seem to remember it.
Suppose I have a nonempty lightface Borel set $X\subseteq 2^\omega$. What is something we can say about how simple some member of $X$ must be?
For example, if $X$ is lightface $\Pi^0_1$, then by the low basis theorem $X$ has a low element, and if $X$ is lightface $\Sigma^0_1$, $X$ has a computable element by obviousness.
My recollection is that every nonempty lightface Borel subset of $2^\omega$ has a hyperarithmetic member, but I can't seem to prove this myself or find this result in Moschovakis' book, so I'm beginning to suspect I'm wrong.
(I'd also like to know - if my recollection is correct - when was this proved, and by whom?)