In his novel "Nausea", J-P. Sartre quotes two or three lines of a gramophone song which has the magic power to dispel nausea: "Some of these days you'll miss me honey" and "When the low moon begins to beam / Every night I dream a little dream". What song is this and where can I hear it?

Two different songs-

"Some of these days you'll miss me honey" is from: Some Of These Days
Words: Shelton Brooks
Music: Shelton Brooks
Performed By: Sophie Tucker (1911), Ella Fitzgerald
-from https://genius.com/Sophie-tucker-some-of-these-days-lyrics listen at http://www.youtube.com/watch?v=oFrGiEUNTkI

"When the [mel]low moon begins to beam / Every night I dream a little dream". is from: THE MAN I LOVE
From the Broadway Musical "Lady, Be Good" (1928)
(George & Ira Gershwin) http://lyricsplayground.com/alpha/songs/t/themanilove.shtml listen at http://www.youtube.com/watch?v=h6ExyYYewV0 (note that most recordings leave out the first verse with the lyrics you seek)

It would appear that J-P Sartre was a Sophie Tucker fan!