The answer is there is no single father of rational choice theory. The reason is that rational choice theory is not so much a theory but rather a coherent framework (others would call it a paradigm) resting on the key foundations of methodological individualism and instrumental rationality / self interest.
Both of these aspects played important roles in the marginal revolution in the late 19th century. Important names here are, of course, Walras, Jevons, Menger and Marshall although none of them can be regarded single father of Economics in the modern sense.
Some decades later, Robbins (1932, p. 15) famously defined Economics as
the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses.
This definition nicely summarizes much of the framework rational choice theory is embedded in. It contains instrumental rationality, in so far as that humans have goals (denoted as preferences) but only limited resources (constraints) and they choose so as to maximize their utility. It also hints at methodological individualism because it places human behaviour at the center of its investigation. Note, however, that Robbins again only summarized a research paradigm that was already well developed. (He says so himself on the same page.)
Still, I guess it is fair to say that rational choice theory as we know it today took off after WW2. Here different names come to mind, depending on which focus you want to have. Arrow/Debreu for general equilibrium theory, von Neumann/Morgenstern if you want to stress expected utility (and game theory). A bit later, Becker is a good choice if you want to stress that rational choice theory is an approach that is driven by its methodology (rational behavior, methodological individualism) and need not investigate an economic topic at all.
But yeah, asking for a single father of rational choice theory would really stress things a bit too far.