**Update.** (June, 2017) François Dorais and I have completed a paper that ultimately grew out of this questions and several follow-up questions. 

> F. G. Dorais and J. D. Hamkins, [When does every definable nonempty set have a definable element?](http://jdh.hamkins.org/when-does-every-definable-nonempty-set-have-a-definable-element/) ([arχiv:1706.07285](https://arxiv.org/abs/1706.07285))
> <blockquote><em><strong>Abstract.</strong> The assertion that every definable set has a definable element is equivalent over ZF to the principle $V=\newcommand\HOD{\text{HOD}}\HOD$, and indeed, we prove, so is the assertion merely that every $\Pi_2$-definable set has an ordinal-definable element. Meanwhile, every model of ZFC has a forcing extension satisfying $V\neq\HOD$ in which every $\Sigma_2$-definable set has an ordinal-definable element. Similar results hold for $\HOD(\mathbb{R})$ and $\HOD(\text{Ord}^\omega)$ and other natural instances of $\HOD(X)$.</em></blockquote>
> Read more [at the blog post](http://jdh.hamkins.org/when-does-every-definable-nonempty-set-have-a-definable-element/).

Click on the history to see the original answer. Related questions and answers appear at:
<ul>
 	<li><a href="https://mathoverflow.net/q/180850/1946">Can $V\neq\HOD$, if every $\Sigma_2$-definable set has an ordinal-definable element?</a></li>
 	<li><a href="https://mathoverflow.net/q/180734/1946">Is it consistent with ZFC (or ZF) that every definable family of sets has at least one definable member?</a></li>
 	<li>We also make use of the $\Sigma_2$ conception of <a href="https://jdh.hamkins.org/local-properties-in-set-theory">Local properties in set theory</a>.</li>
</ul>