According to the following paper, this invariant's introduction is sourced as: A. Casson, Lecture notes, MSRI Lectures, Berkeley, 1985.
The first published discussion appears to be found in: S. Akbulut & J. McCarthy, Casson's invariant for oriented homology 3-spheres, an exposition, Math. Notes, No. 36, Princeton Univ. Press, Princeton, 1990.
"I don't know if he [Casson] has ever written his notes. I attended his lectures at MSRI then I went back to MSU and tried to fill in the details together with McCarthy, those notes are the result." S. Akbulut
"Here is a pdf of hand-written notes of T. Cochran's. Page two includes annotated notes by Boyer, and I got the notes from my friend and colleague, A. Nicas." H. Boden