I'm looking for some form of the Recursion Theorem which could motivate the existence of the recursively defined function $F \left ( f \right )$ below.
For any set $A$ define the set of ordered tuples (or ordered lists in the computer science sense) with entries from the set $A$. For any set $A$ \begin{align*} L_0^A &= \left \{ \emptyset \right \} \\ L_n^A &= \left \{ (x, y): x \in A, \, y \in L_{n-1} \right \} \\ L(A) &= \bigcup \left \{ L_n^A : n \in \mathbb N \right \} \end{align*}
Then given $f: A \to B$ I want to motivate the existence of the function $F \left (f \right ): L(A) \to L(B)$ defined by the recursion $$ F(f)(a) = \begin{cases} \emptyset & \text{if } a = \emptyset \\ \left ( f(x), F\left (f \right )(y) \right ) & \text{else} \end{cases} $$
Do you have a reference to a form of the Recursion Theorem that could be used and a quick pointer to how?
Here is a proof using Recursion Theorem as in the wikipedia link. Let $X$ denote the set of all partial functions from $L(A)$ to $L(B)$. We construct a function $$G:\mathbb N\to X\text,$$ such that $G(n)$ is a function $L^A_n\to L^B_n$, as follows:
Define $G_0:L^A_0\to L^B_0$ as the empty function. Define $g:X\to X$ as $$ g(h)=a \mapsto \begin{cases} h(a) &\text{ if } a\in\mathrm{dom}(h) \\ (f(x), h(f(y))) &\text{ if } a=(x,y) \text{ and }y\in\mathrm{dom}(h) \\ \varnothing &\text{ otherwise } \end{cases} $$ By recursion theorem, one acquire a function $G$ with $G(0)=G_0$ and $G(n)=g(G(n-1))\restriction L^A_n$. Now, one easily realize that when $a=(x,y)\in L^A_n$, we have $y\in \mathrm{dom}(G(n-1))$ and thus $G(n)(a)=(f(x),G(n-1)(y))$. Also, $G(m)$ coincide with $G(n)$ on the intersection of their domain.
Finally, let $F=\bigcup_n G(n)$. $F$ has the expected properties.
However, I believe a direct application of Recursion Theorem like this is lengthy and tedious, if not pedantic. In mathematics $L(A)$ is a well-founded relation, so one can just do recursion on it. This simplify the process to merely one step. You can find more information on Theorem I.9.11 of Kunen's Set Theory (2011 Edition), or this webpage.