There are a few ways I think of averages of a (discrete) random variable:
Average is an OLS estimate of running a regression of the random variable on a constant term. In that sense, it is a value that 'best' represents the data (minimizes the Euclidean distance)
A central tendency of the data (i.e. my best guess of the random variable, without having any data on it)
Now, in many cases, the average is not included in the support of the random variable. For instance, the expected value of the outcome of rolling a die is 3.5. However, this value is not included in the support. How would one interpret the average in this case?