Todd asked “Why [did Church choose] lambda and not some other Greek letter?”. Here are three answers:
The story is that in the 10s and 20s, mathematicians and logicians used ^ as a notation for set abstraction, as in ^i : i is prime. Church used ^` (i.e., a primed version of this symbol) for function abstraction, because functions are just sets with extra properties. The first type setter/secretary read it as λ and Church was fine with. True or not? I don’t know but it’s fun.
(By the way, why did Church choose the notation “λ”? In [Church, 1964, §2] he stated clearly that it came from the notation “xˆ” used for class-abstraction by Whitehead and Russell, by first modifying “xˆ” to “ˆx” to distinguish function abstraction from class-abstraction, and then changing “ˆ” to “λ” for ease of printing. This origin was also reported in [Rosser, 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 “λ” just happened to be chosen.)
We end this introduction by telling what seems to be the story how the letter ‘λ’ was chosen to denote function abstraction. In  Principia Mathematica the notation for the function f with f(x) = 2x + 1 is 2xˆ +1. Church originally intended to use the notation xˆ .2x+1. The typesetter could not position the hat on top of the x and placed it in front of it, resulting in ˆx.2x + 1. Then another typesetter changed it into λx.2x + 1.