Chapter 12 of Jech & Hrbacek's "Introduction to Set Theory" asks the following as exercise 5.4:
Assuming $\diamondsuit$ show that there exists $2^{\aleph_1}$ stationary subsets of $\omega_1$ such that the intersection of any two is at most countable.
[Hint: Consider $S_X = \{ \alpha < \omega_1 \space|\space X \cap \alpha \in W_\alpha\}$ for all $X \subset \omega_1$.]
$\diamondsuit$ is Principle Diamond, for which I believe the book uses a weaker version than that used by other books I looked up, hence I quote from page 234:
Principle $\diamondsuit$ $\quad$ There exists a sequence $\langle W_\alpha \space | \space \alpha < \omega_1\rangle$ such that, for each $\alpha < \omega_1$, $W_\alpha \subset P(\alpha)$, $W_\alpha$ is at most countable, and for any $X\subset \omega_1$ the set $\{\alpha < \omega_1 \space | \space X \cap \alpha \in W_\alpha\}$ is stationary.
Usually the exercises from this book are easy, and most of the time the hints give out almost the entire answer. That is why I suspect the answer to be a subset of the set $S$ that contains all the sets of the form that the hint suggests. But, my problems are:
1) Given $X\subset\omega_1$, how can I choose a $Y\subset\omega_1$ such that the intersection of $S_X \cap S_Y$ is at most countable?
2) It is easy, given $X\subset\omega_1$, to choose $Y\subset\omega_1$ such that $S_X \neq S_Y$. I can even go further and construct a countable subset of $S$, but I cannot seem to construct or find one of size $2^{\aleph_1}$. How can I guarantee a subset of that size?