Why did Alonzo Church choose the letter $\lambda$ as the "binding operator"? Is there any known reason why Alonzo Church chose Greek $\lambda$ as the "binding operator" for the Lambda Calculus?
 A: 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 Unpublished letter to Harald Dickson, §2]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 $\psi\hat z$.”
  [...]

