Why is variance squared rather than cubed (or any other exponent)? Variance is

$$\dfrac{\sum_{i=1}^{n}(x_i-\bar x)^2}{n-1}$$

But why square the difference? Why not cube it, or any other exponent?

Related question.
You can also use other exponents, but then it is not called variance anymore. See this wikipedia page, for example.

Variance comes up quite naturally in various contexts, and carries a lot of information about the distribution.
Still, I am not convinced by this answer. 
The naming convention can be separate topics to discuss.
Here what is the issue if we take cubes or quad etc?