What is it meant by a "hybrid argument"? Can anyone explain (or point to a reference for) what a hybrid argument is in a security proof, and when it's convenient or preferable to use it?

Among some of the places where I've seen it mentioned, there is the paper Boneh, Sahai, Waters - Functional Encryption: Definitions and Challenges. The term standard hybrid argument is mentioned in the proof sketch of Theorem 1 within section 4.1.
Quoting from "On beating the hybrid argument" (by Bill Fefferman, Ronen Shaltiel, Christopher Umans and Emanuele Viola; 2012):


  The hybrid argument allows one to relate the distinguishability of a distribution (from uniform) to the predictability of individual bits given a prefix. The argument incurs a loss of a factor $k$ equal to the bit-length of the distributions: $\epsilon$-distinguishability implies $\epsilon/k$-predictability. 


As you're looking for a more in-deep and more practical explanation of "hybrid argument", I would like to recommend diving into the PDF "The hybrid argument", which includes a whole bunch of examples to walk through so you can wrap your head around it completely.