There seems little dispute that Tennessee Williams coined the phrase "memory play" to describe his play The Glass Menagerie, in which one of the characters is also the narrator of events drawn from their own memory. He uses it in the play's opening soliloquy.
The play is memory. Being a memory play, it is dimly lighted, it is sentimental, it is not realistic. In memory everything seems to happen to music. That explains the fiddle in the wings. I am the narrator of the play, and also a character in it. The other characters are my mother Amanda, my sister Laura and a gentleman caller who appears in the final scenes.
However, while Williams may have invented the phrase I am not so certain that he invented this concept, and that other plays based on the same framing device may have existed beforehand. I haven't been able to find anything definitive, as links point back to The Glass Menagerie. Is it truly the original memory play, or was there an earlier forebear?