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 “$\hat x$.”

A preprint of that article is available online, here is a quote from it:

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 Unpublished letter to Harald Dickson, §2]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.