It seems like Gödel didn't use the letter $L$ for his model before his book "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory", which is probably the first place it got used.
Do anyone of you know why he used the letter $L$? It does seem like a bit ad hoc in the book, where he names some functions $J_i$, then some other functions $K_i$ and then ends up defining $L$. But why $J_i$ then?
(I'm sorry if this is not suited for MO)