Reading the above, I disagree with a few. For instance:
Fraulein Schneider has at least one more solo and one duet; Lotte Lenya was nominated for Best Actress in a Musical, so I would not consider that meeting the criteria set.
I would also question whether Carlotta is an insignificant character in Follies. Most of the roles in Follies are modest. I would agree with the poster re a number of the songs in Follies probably being reasonable proposals.
I don't disagree re The Miller's Son, although I have never understood the appeal of that song. I have always thought of it as filler, even though it has 'an' 11:00 o'clock slot in the show.
I would probably think of a several numbers from Nine, with Be Italian being the best 'match'. I could also argue that Lilliane had very little to do other than number, for which Liliane Montevecchi won a Tony.
It probably doesn't quite meet the bill, but the lion's share of I Have Dreamed is sung by the male, who is barely in the show.
Also maybe a little stretch, but Sit Down, You're Rocking the Boat comes close.
Another popular number from Gypsy whose charms have always escaped me is All I Need now Is the Girl.
Don't know if it qualifies as a great song, but The Grass is Always Greener seems to fit. Marilyn Cooper stopped the show and won a Tony for a role that probably involved 6 minutes of stage time.
I don't think any of the songs from Whorehouse remotely meet the 'great songs' requirement.
Agree re Not a Day Goes By, simply because of the timing of the song. Someone we have not seen previously gets arguably the best song in the show. Her role is a little larger, but when she actually sings the song, she is insignigicant. I remember being surprised at how little applause the number received relative to others in the show, but concluded that was because we had no emotional connection to the character, which is a (minor) gotcha with Merrily's structure.