As Henning said in a comment, from the context it must mean the function $z\mapsto z$, although it didn't quite say as much. This function "canonically" generates the polynomial functions, and $A$ is the closure of the polynomials in the sup norm on the closed disk. In this context, "canonical" seems to mean, "what you would expect; no tricks here."
For reasons I cannot fully explain, this function is often named awkwardly. It is sometimes tempting to call it "the identity function" but this is not an ideal way to think of the function, given that the algebraic operations in $A$ are pointwise operations with values in $\mathbb C$ (rather than, say, composition of self-maps of the disk). It is more like "inclusion," but that isn't quite right, either, in this context.
I would guess that at some level most people just think of this as "the function $z$," or just "$z$" when the context seems clear, just as we would speak of "(the function) $z$ squared" or "(the function) $e$ to the $z$". But particularly in the case of $z$ that has the potential to be confusing or ambiguous, and also might not emphasize enough the role as an element of an algebra, where we sometimes don't want to think of it explicitly as a function. Murphy might have been trying to concisely both avoid ambiguity (not entirely successfully) and emphasize the status of $z$ as generator, as the later plays a role in the application of Theorem 1.3.7 (I presume, although I don't have the book).
I have seen another functional analysis textbook use the phrase "the current variable" in a similar context.
When dealing with subsets of $\mathbb C^n$, the function $(z_1,z_2,\ldots,z_n)\mapsto z_k$ is sometimes called the $k^\text{th}$ "coordinate function." When $n=1$, the map $z\mapsto z$ can be called the coordinate function, too, and I have also seen this used in similar contexts.