Why are rational functions called "rational"?

Because of their similarity to rational numbers; i.e.---

A function $f(x)$ is called a rational function provided that

$$f(x) = \frac{P(x)}{Q(x)},$$

where $P(x)$ and $Q(x)$ are polynomials and $Q(x)$ is not the zero polynomial; just like a real number $n$ is called a rational number, provided that

$$ n = \frac{p}{q},$$

where $p, q$ are integers and $q \neq 0$.

a rational < insert thing > is pertaining to a ratio. A rational number is a ratio of integers, a rational expression is a ratio of expressions,
and a rational function is a function whose result comes from evaluating a rational expression.

As the other answers suggest, "rational" here means "being a ratio". However, of course you could write everything, say $x$, as a ratio, namely as $\dfrac{x}{1}$.

_{(OK my very precise math friends, not "everything", but everything that's contained in some monoid with $1$; certainly everything that would usually be called a "number" or a "function".)}

So the actual question is: What are the "more elementary elements" (numbers or functions) of which the "rational" numbers, or functions, are ratios?

It turns out that we call

rational numbers = ratios (quotients) of integers

rational functions = ratios (quotients) of polynomials.

So there seems to be a deeper analogy hidden here, that is

Polynomials are among functions what integers are among numbers; polynomials are "the integers among the functions";

and that is true to a surprisingly large extent.