How to prove a language isn't necessarily regular?
If we take $L = \{a^n \mid n \in \mathbb{N}\}$, then $L' = \{a^{2n} \mid n \in \mathbb{N}\}$, which is a perfectly fine regular language. Or even more trival, we could observe that $\emptyset' = \emptyset$.
As such, it does not hold that $L'$ is never regular for a regular language $L$. This means that answering the question will involve constructing a particular language $L$ for which you can prove that $L'$ is not regular (and such a proof could use the pumping lemma).
As a heuristic for how to attempt this, I would first look at some typical examples of non-regular languages, and try to find a way to express them as $L'$ for some regular language.  If I don't succeed with that, I'd attempt to built $L$ by simultaneously ensuring that $L$ itself is regular, and by running the pumping lemma on $L'$ to prove that it is not.