This question has been answered on math.SE (as pointed out by Joel David Hamkins). With a reference to Lambda-Calculus and Combinators in the 20th Century by Felice Cardone and J. Roger Hindley, Handbook of the History of Logic Volume 5, 2009, Pages 723–817, it is stated that “$\lambda x$” comes from “$\hat x$” in Principia Mathematica. Here is a quote from a preprint of Lambda-Calculus and Combinators in the 20th Century:
By the way, why did Church choose the notation “$\lambda$”? In [A. Church, 7 July 1964. Unpublished letter to Harald Dickson, §2] he stated clearly that it came from the notation “$\hat x$” used for class-abstraction by Whitehead and Russell, by first modifying “$\hat x$” to “$\wedge x$” to distinguish function-abstraction from class-abstraction, and then changing “$\wedge$” to “$\lambda$” for ease of printing. This origin was also reported in [J. B. Rosser. Highlights of the history of the lambda calculus. Annals of the History of Computing, 6:337—349, 1984, p.338]. On the other hand, in his later years Church told two enquirers that the choice was more accidental: a symbol was needed and “$\lambda$” just happened to be chosen.
Assuming that “$\lambda x$” comes from “$\hat x$” in Principia Mathematica, let us look how it is used there.
In Principia Mathematica there are two ways the notation “$\hat x$” is used. The first use is to write “propositional functions,” it is introduced in Volume I, in Chapter I of the Introduction, on page 15. Here is a quote:
[...] When we wish to speak of the propositional function corresponding to “$x$ is hurt,” we shall write “$\hat x$ is hurt.” Thus “$\hat x$ is hurt” is the propositional function, and “$x$ is hurt” is an ambiguous value of that function. Accordingly though “$x$ is hurt” and “$y$ is hurt” occurring in the same context can be distinguished, “$\hat x$ is hurt” and “$\hat y$ is hurt” convey no distinction of meaning at all. [...]
The second use is to write classes in a way similar to the modern “$\{\,z\mid\psi(z)\,\}$”, it is introduced in Volume I, in Section C of Part I, in definition *20.01, on page 197. Here is some quote:
[...] But it is convenient to regard $f\{\hat z(\psi z)\}$ as though it had an argument $\hat z(\psi z)$, which we will call “the class determined by the function $\psi\hat z$.” [...]
