Most of the notations in Artificial Intelligence are borrowed from the mathematics.
$x$ stands for input (vector), $y$ stands for output (vector) etc., and the list is long.
But, I am not sure whether $z$ has any (widely used) role in mathematics.
Is there any reason behind the usage of letter $z$ to represent a noise vector? Or is it just selected randomly without any reason?
I don't think there's any rationale behind the usage of the letter $z$ to denote the noise (which sometimes is also denoted by $\epsilon$ in other contexts), apart from the fact that $x$ and $y$ are already being used and that the letters $x$, $y$, $z$ and $w$ are often used to denote variables in mathematics. In particular, in machine learning, $x$ and $y$ are often used to denote the inputs and outputs (or labels) respectively, while $w$ often denotes the parameters (although $\theta$ is also used for that).
In other words, it's just a convention (e.g. $z$ is also used to denote the hidden variable in the VAE paper). Even if it wasn't and someone used $z$ as a mnemonic letter or for some particular reason, I wouldn't lose too much time on this issue, as another author may very well use any other letter to refer to the same concept. It's important to be a bit flexible when it comes to notation in mathematics, otherwise, you may easily get lost.
