What would one call the mean of the mean and median? If we wanted to take the mean of the mean and median, what would we call that value? 
I'm considering this as a useful summary statistic with a heavily skewed distribution.
For instance, if the mean of a distribution is 10 and the median is 15, the mean of these two values is 12.5.
 A: Apparently, you want to use this measure of centrality in hopes
that it will somehow be a better summary or descriptive static
than either the mean or the median. I have not seen it used, so
I won't speculate whether you will find it useful.
I suggest that you consider using a 'trimmed mean' which has been
much studied as a 'robust statistical measure' of centrality, and has been
widely used in some areas such as 'exploratory data analysis.'
(You can google the terms enclosed in single quotes for more information.)
A 5% trimmed mean of a sample is the mean of the observations that
remain after the top 5% and the bottom 5% have been discarded; roughly
that is the mean of the middle 90% of the data. (Conventions have to be
established for what to do when "5%" of the data isn't an integer,
and there is not much point in using a 5% trimmed mean for a sample
size much smaller than $n = 20.$) 
Percentages other than 5% are also used with the obvious modification.
Essentially, a median is a 50% trimmed mean.
Here is an example of 100 observations from an exponential population
with rate .01 (population mean 100 and population median 69.31). This is a severely right-skewed distribution,
so usually the median is smaller than the 5% trimmed mean, which is
usually smaller than the (ordinary) mean. These are marked as red, blue,
and green lines respectively on the stripchart below. In R statistical
software, one uses the statement mean(x, tr=.05) to find the 5% trimmed
mean of the vector x of observations. (The sample median, 5% trimmed mean,
and mean are shown in the output below.)
x = rexp(100, .01)
stripchart(x, pch=20)
h = median(x);  t = mean(x, tr=.05);  a = mean(x)
h; t; a
## 75.08734   # sample median
## 88.08616   # sample 5% trimmed mean
## 96.90833   # sample mean
abline(v = c(h, t, a), col=c("red", "blue", "green"), lwd=2)


Here are similar stripcharts for two additional random samples of size $n = 100$
from the same population. For such a small sample, one cannot expect
sample means and medians to be really accurate estimates of the population mean and median.
h; t; a
## 76.65084
## 89.44892
## 95.51764


h; t; a
## 69.14684
## 92.0044
## 104.1352


The sample 5% trimmed mean is obviously not exactly the same thing as the
average of the sample mean and median. But it seems to behave similarly,
and it has the advantage of having been studied. And various trimmed means
have been widely used applications (including Olympic scoring and government economic indexes).
