Why is lambda calculus named after that specific Greek letter? Why not “rho calculus”, for example? Where does the choice of the Greek letter $\lambda$ in the name of “lambda calculus” come from? Why isn't it, for example, “rho calculus”?

The symbol “λ” is used for one of two basic constructions in the system introduced by Alonzo Church, specifically abstraction. The notation did not just happen to be chosen but was to distinguish it from another construction by Whitehead and Russell represented as “xˆ.” For his new system, Church initially used “∧x,” then replaced it to “λx” to ease printing, obviously, interpreting the former logical symbol as the capital Greek letter “Λ.”
See “History of λ-calculus and Combinatory Logic” by J. R. Hindley, F. Cardone (Handbook of the History of Logic, 5: 723–817, Elsevier, 2009).
