To answer, the assertion that volatility is easier to predict than expected return requires clarification. The phrase "easier to predict" is particularly ambiguous.
To me this means that the estimation of volatility from a sample of returns is more robust than the estimation of expected return in the context of relative sampling error.
Suppose over a time period $T$ we observe asset prices $S_0,S_1, \ldots, S_N$ at uniformly spaced time intervals of length $\delta t$ where $T = N \delta t$. Assume that the log-return (over an interval of length $\delta t$) has a stable distribution and returns over non-overlapping intervals are independent. Let $\mu$ and $\sigma$ denote the annualized mean return and volatility, respectively.
The $\delta t$-period log-return has expected value $\mu \delta t$ and variance $\sigma^2 \delta t$, where the $\delta t$ scaling of the variance is a consequence of the independence. We now have an iid sample $X_1,X_2,\ldots, X_N$ where
$$X_j = \log \frac{S_j}{S_{j-1}}$$
and the estimators for expected retrun and volatility are
$$\hat{\mu}\delta t = \frac{1}{N}\sum_{j=1}^N X_j, \quad \hat{\sigma}^2\delta t = \frac{1}{N-1}\sum_{j=1}^N (X_j - \hat{\mu}\delta t)^2$$
Asymptotically, the sampling distributions for the estimators are
$$\hat{\mu}\delta t \sim \mathcal{N}(\mu \delta t, \sigma^2 \delta t/N),\quad \frac{(N-1) \hat{\sigma}^2 \delta t}{\sigma^2 \delta t} \sim \chi^2(N-1),$$
that is, normal and chi-square with $N-1$ degrees of freedom, respectively. The standard errors for the estimates of expected return and volatility are, respectively, $\sigma\sqrt{\frac{\delta t}{N}}$ and $\frac{\sqrt{2} \sigma^2 \delta t}{\sqrt{N-1}}$.
As expected, the absolute sampling error (given by standard error) for the both the expected return and the volatility diminish as $1/\sqrt{N}$ as the number of samples $N$ increases.
However, the relative errors tell a different story. The relative sampling error for the volatility is
$$\frac{\frac{\sqrt{2} \sigma^2 \delta t}{\sqrt{N-1}}}{\sigma^2 \delta t} = \sqrt{\frac{2}{N-1}}$$
This shows that the relative error improves simply by increasing the number of samples. Given a fixed time period $T$, we only need to sample returns at a higher frequency to improve the estimate of volatility. Sampling daily is more accurate than sampling monthly, sampling monthly is more accurate than sampling quarterly, etc.
On the other hand, the relative sampling error for the expected return is
$$\frac{\sigma \sqrt{\frac{\delta t}{N}}}{\mu \delta t} = \frac{\sigma}{\mu \sqrt{N \delta t}}= \frac{\sigma}{\mu \sqrt{T}}$$
The only way to get a better estimate for expected return is to increase the length of the period $T$ over which the samples are observed. For a fixed period $T$, say 3 years, the relative error cannot be improved by increasing the sampling frequency, regardless of how many additional samples are taken. In other words, in order to improve the accuracy of the estimated return by a factor of 5, we must increase the sampling period by a factor of 25 to 75 years -- clearly problematic.
The root cause of this phenomenon would seem to be the fact that return scales like $\delta t$ and volatility, with independent returns, scales like $\sqrt{\delta t}$ with respect to the measurement period $\delta t$.
