Joel gave one generalization; there is another generalization to a different direction:

Let $T$ be a $\xi$-stable (complete) first order theory, and $A$ a set in a model of $T$ with $|A|\le\xi$. If $I$ has cardinality $>\xi$, then $I$ has a subset with cardinality $>\xi$, which is an indiscernible set over $A$. (Here $I$ is a set of finite tuples in some model of $T$ that contains $A$.)

The usual $\Delta$-lemma follows from this because the theory of infinite set with empty language is $\omega$-stable, and an indiscernible set in such a model is just as wanted.

I believe this is from Saharon Shelah's Classification Theory (1990).

Let $T$ be a $\xi$-stable (complete) first order theory, and $A$ a set in a model of $T$ with $|A|\le\xi$. If $I$ has cardinality $>\xi$, then $I$ has a subset with cardinality $>\xi$, which is an indiscernible set over $A$. (Here $I$ is a set of finite tuples in some model of $T$ that contains $A$.)
The usual $\Delta$-lemma follows from this because the theory of infinite set with empty language is $\omega$-stable, and an indiscernible set in such a model is just as wanted.