In a recent CACM article , the authors present an implementation for staged functions. They use the term as if it was well-known, and none of the references looks like an obvious introduction.
They give a short explanation (emphasis mine and reference number changed; it's 22 in the original)
In the context of program generation, multistage programming (MSP, staging for short) as established by Taha and Sheard  allows programmers to explicitly delay evaluation of a program expression to a later stage (thus, staging an expression). The present stage effectively acts as a code generator that composes (and possibly executes) the program of the next stage.
However, Taha and Sheard write (emphasis mine):
A multi-stage program is one that involves the generation, compilation, and execution of code, all inside the same process. Multi-stage languages express multi-stage programs. Staging, and consequently multi-stage programming, address the need for general purpose solutions which do not pay run-time interpretive overheads.
They than go on to several references to older work allegedly showing that staging is effective, which suggests that the concept is even older. They don't give a reference for the term itself.
These statements seem to be orthogonal, if not contradictory; maybe what Rompf and Odersky write is an application of what Taha and Sheard propose, but maybe it is another perspective on the same thing. They seem to agree that an important point is that programs (re)write parts of themselves at runtime, but I do not know whether that is a necessary and/or sufficient ability.
So, what is staging respectively are interpretations of staging in this context? Where does the term come from?
- Lightweight Modular Staging: A Pragmatic Approach to Runtime Code Generation and Compiled DSLs by T. Rompf and M. Odersky (2012)
- MetaML and multi-stageprogramming with explicit annotations by W. Taha and T. Sheard (2000)