  A Steinhaus random multiplicative function $f$ is a completely multiplicative
function obtained by setting its values on primes $f(p)$ to be independent
random variables distributed uniformly on the unit circle. Recent work of
Harper shows that $\sum_{n\le N} f(n)$ exhibits ``more than square-root
cancellation," and in particular $\frac 1{\sqrt{N}} \sum_{n\le N} f(n)$ does
not have a (complex) Gaussian distribution. This paper studies $\sum_{n\in
{\mathcal A}} f(n)$, where ${\mathcal A}$ is a subset of the integers in
$[1,N]$, and produces several new examples of sets ${\mathcal A}$ where a
central limit theorem can be established. We also consider more general sums
such as $\sum_{n\le N} f(n) e^{2\pi i n\theta}$, where we show that a central
limit theorem holds for any irrational $\theta$ that does not have extremely
good Diophantine approximations.
