For 'statistical properties' read 'computing the distribution of', or 'computing some aspect of the distribution of' (such as variance, say).
In particular, they're probably referring to the sampling distribution of a sample statistic.
Sample means have some quite nice properties*, so that in many cases, for example, it's relatively easy to compute the mean, variance (and covariances) of the distribution of sample means, and asymptotically, we have the central limit theorem which (speaking very loosely) tells us about distributions of means in large samples.
* means of sums "add", and variances of sums of independent variables also "add" (i.e. the mean of a sum is the sum of the means, the variance of a sum of independent variables is a sum of the variances, which makes the mean and variance of the distributions of sample means - usually - quite easy to find.)
By contrast, sample medians (and other quantiles) are often more difficult to work with, and don't have nice linear properties like that. Nevertheless, sometimes we can make progress in finite samples, and asymptotically (i.e. in very large samples) they tend to behave relatively more 'nicely' - there's a nice result about non-extreme quantiles (in effect, a kind of central limit theorem) and a different kind of theorem relating to extremes.
Modes are much worse. Generally speaking, they really don't have very 'nice' properties; for example, it's relatively easy for modes to 'jump about' in somewhat surprising ways when you take larger samples, and even asymptotically the variance of a mode doesn't decrease as $1/n$, so they may tend to be pretty 'noisy' things to deal with. Even the basic step of defining a sample mode for continuous data can be a tricky business.