The mean of a collection of random variables is itself a random variable.
So if you have say, $X_1,X_2,X_3,...,X_n$, then $\bar{X}=\frac{1}{n}(X_1+X_2+X_3+...+X_n)$ is a random variable.
If you're talking about a sample of particular observations, $x_1, x_2, ..., x_n$, then you could look at the sample mean, $\bar{x}$ as another kind of observation, relating to the random variable $\bar{X}$ in the same way that the observation $x_1$ relates to the random variable $X_1$, which is to say it's a realization of the random variable $\bar{X}$.
[On the other hand, when we're talking about the mean of the distribution of a random variable, that is a parameter; it's not a variable $-$ at least not in a frequentist framework; it may be treated as one in a Bayesian framework, in which case it can be taken to represent the uncertainty in our belief/knowledge about its value.]