EDIT: I now see that the question was asked and answered more directly, but it's not clear to me that that answer makes sense and it will take me a while to parse.
In Kunen's Set Theory, chapter II question 51, we have a couple equivalent statements to $\diamondsuit$ for which it seems natural to directly prove their equivalence before moving to the others (and I have made good progress there). This question has already been asked, but not answered in a way that's useful for a person as stumped as I. Quoting the original:
There exists a sequence $\langle f_\alpha:\ \alpha<\omega_1\rangle$ of functions $f_\alpha:\alpha\to\alpha$ such that for all functions $f:\omega_1\to\omega_1$ the set $\{\alpha<\omega_1:\ f|_\alpha=f_\alpha\}$ is stationary in $\omega_1$.
and:
There exists a sequence $\langle f_\alpha:\ \alpha<\omega_1\rangle$ of functions $f_\alpha:\alpha\to\alpha$ such that for all functions $f:\omega_1\to\omega_1$ there is $\alpha>0$ such that $f|_\alpha=f_\alpha$.
As before, the first clearly implies the second because stationary sets are nonempty. On the other hand, I am completely lost trying to go in the other direction (and the other equivalents are seemingly further away).
I'm having a lot of trouble finding any path to get to stationary; we know from the book that $\omega_1$ can be partitioned into $\omega_1$ disjoint stationary sets, but that doesn't seem too applicable for getting stationary agreement with all $f:\omega_1\to\omega_1$.
Fodor's lemma seems like it ought to be helpful, but again I am not sure how to start with one (or many?) useful stationary set(s) or what sort of decreasing function addresses the issue.
I have also considered a tree perspective, where if we put all these functions in the tree $\omega_1^{\omega_1}$ where higher functions extend lower ones, then $\langle f_\alpha:\ \alpha<\omega_1\rangle$ contains a maximal antichain. Perhaps there's some way to build another $\omega_1$ maximal antichains of size at most $\omega_1$ with small intersection that, together, have stationary intersection with each maximal chain, but again I have no idea how to get up to stationary.
I would be quite thankful to anyone for a key hint! This one has confounded me.