A slightly obtuse way to say that a first-order theory $T$ has Skolem functions is to say that for any $M\models T$ and $A\subseteq M$, $\mathrm{dcl}(A)\preceq M$.
This suggests a similar condition with algebraic closure replacing definable closure, namely the condition that for any $M \models T$ and $A \subseteq M$, $\mathrm{acl}(A)\preceq M$. I'm sure this condition has a name and I feel like I've seen it before, but I couldn't find it, so I'll just call this property $(\ast)$.
In a paper of Kruckman and Ramsey they showed that under certain conditions it's possible to Skolemize an $\mathrm{NSOP}_1$ theory without breaking $\mathrm{NSOP}_1$. They also allude to a result of Nübling that makes it seem unlikely you can do much better.
In an extremely nice case, namely that of strongly minimal theories, it's always possible to expand the theory to satisfy $(\ast)$ without breaking strong minimality, specifically by adding constants naming the elements of the prime model.
So a specific initial question would be this:
Given a stable theory $T$ can you always expand $T$ to satisfy $(\ast)$ while preserving stability? If $T$ is $\kappa$-stable can you ensure that the expansion is also $\kappa$-stable?
And then of course you can ask similar questions about other neo-stability theoretic dividing lines, such as $\mathrm{NIP}$.
