When is it a good idea to just use the average for imputation? Suppose we have a data set test:
1
8
12
14
.
.
19

The  .  denotes missing values. When would it be better to use the average of the non-missing values to impute the missing values rather than assuming that the data comes from a normal distribution?
 A: Don Rubin wrote an influential paper proving that there is no single imputation method that will produce unbiased inferences (where "single imputation" means the imputing of only one value for a missing observation). However, in the same paper he pointed out that it may well be possible to create multiple imputations whose mean is an unbiased estimate of the missing value, and whose contributions to increased variance in subsequent analysis is a reasonable estimate of the added uncertainty due to data missingness (and therefore provides a reasonable route to estimating standard errors, p-values, confidence intervals, etc.).
This is his paper:
Rubin, D. B. (1976). Inference and missing data. Biometrika, 63(3):581–592.
And this an update to it:
Rubin, D. B. (1996). Multiple imputation after 18+ years. Journal of the American Statistical Association, 91(434):473–489.
And this a gentle introduction to the topic of multiple imputation:
Schafer, J. L. (1999). Multiple imputation: a primer. Statistical Methods in Medical Research, 8:3–15.
There are a variety of statistical software packages that support multiple imputation (e.g. mice in R, or ice in Stata, or indeed Stata's built-in multiple imputation capabilities in recent versions).
A: It is never a good idea to do this, but, if there is very little missing data then it will do relatively little harm, will be much easier to implement and, depending on your final audience, may be a lot easier to explain.  However, a relatively sophisticated audience may object to the single mean imputation. 
A: The question: 
"What imputation method is the best choice" is alway dependend on the dataset you look at
Taking the mean, in general is a valid imputation method. As somebody already mentioned, it is easy to explain for publications and it has its advantages in computing speed.
Mean as a imputation method is a good choice for series which randomly fluctuate around a certain value/level. 
For the series shown, mean doesn look appropriate.
Since it is also just one variable you cannot use classical multivariate algorithms as provided by mice, Amelia, VIM.
You would have to look especially at time series algorithms.
One simple, yet for your example good like approach would be a linear interpolation.
library(imputeTS)   
x <- c(1,8,12,14,NA,NA,19)
na.interpolation(x)

Here is the output for a linear interpolation:
[1]  1.00000  8.00000 12.00000 14.00000 15.66667 17.33333 19.00000

This is probably a better result than the mean.
There are also more advanced time series methods in the imputeTS package (by me) or one in the forecast package (by Rob Hyndman)
