Q:

Where does the delta method's name come from?

Where does the delta method's name come from?
I don't see anything related to "$\epsilon$-$\delta$", nor Dirac delta for example.

A:

The name "Delta" is from the symbol $\Delta$ for "change" which is used in limit expressions like let $\Delta X_i\rightarrow 0$, where $\Delta X_i=X_{i+1}-X_i$, and also $\Delta$ or lower case "δ" refers to an inexact, non-zero differential equation (i.e., before limits are taken) which reduces in the limits to a differential equation the latter using infinitesimals. For example, $\Delta f(x)= f(x+\Delta x)-f(x)$, which reduces to $\partial f$, a vanishingly small differential, for $\Delta x$ sufficiently small. Now sometimes we want to manipulate equations so that we are not dividing or multiplying by zeros, and in that case, we might use the $\Delta $ notation.
The name delta method, is probably only used by statisticians. In other contexts, e.g., engineering, physics, it would be more commonly referred to as propagation of error. To see how the error propagation includes the $\Delta$ symbol, see propagation of error.
 Edit

Error propagation is from Taylor series, useful examples. Indeed, when extended to the first two derivatives of Taylor series, i.e., $\mathbf{Var}[Y]\approx(g'(\mu_X))^2\sigma_{X}^{2}+g'(\mu_X)g''(\mu_X)\mu_3+\frac{1}{4}(g''(\mu_X))^2(\mu_4-\sigma_{X}^{4})$, it becomes more accurate, and I would venture that that is needed when the solution space curvature is of greater magnitude. 
BTW, in ancient Greek the letter $\Delta$ was pronounced as a "d" sound and in modern Greek this is a hybrid "d" and "th" sound. The etymology is purported to be as follows,
c. 1200, Greek letter shaped like a triangle, equivalent to our "D," the name from Phoenician daleth "tent door." Herodotus used it of the mouth of the Nile, and it was so used in English from 1550s; applied to other river mouths from 1790.

