Q:

What may be an inefficient estimator of the population mean?

If the sample mean is an efficient estimator of the population mean, what may be an example of an inefficient such estimator? 

A:

Consider a sample of size $N$ drawn from a normal distribution. The sample median $\tilde{X}$ is an unbiased and consistent estimator for $\mu$. For large 
$N$ the sample median is approximately normally distributed with mean $\mu$ and variance 
$\pi/2N$. The efficiency for large $N$ is thus $2/\pi \approx 0.64$. This is the asymptotic efficiency, that is the efficiency in the limit as sample size $N$ tends to infinity.
Edit: The sample mean is a better (more efficient) estimator for the population mean as opposed to the sample median, because the sample median will miss the population mean by more (on average) than the sample mean.
That is, the sample medians (talking asymptotically - as the sample size N reaches infinity) will have a wider distribution compared to the sample means. And since both estimators are unbiased, meaning they will estimate the true population mean - meaning they wont under/overestimate the population mean, all that remains is to look at the variance of these estimators. And as written above, the sample median distribution will have a higher variance compared to the sample mean, which means the sample median is a less efficient estimator for the population mean.

