My Name is a bad song? I always kind of liked it - well, I guess it depends on how it's donet. If the guy playing Bill Sykes is mean enough,and the orchestrator doesn't soften up the "clanging" sounds (forgive me for not knowing the musical term) the song definitely can work.
I'm also surprised you'd say A Man Doesn't Know -- not the best lyrics, but the melody is nice -- it's certainly not a "terrible" song.
I would have to go with:
Clambake (I know, but it IS a terrible song)
Something Baaaaaaaaaaaaaaaaaaaad ('nuff said)
That damn song in Evita that I can't think of the name of on the concept recording (and I think the movie soundtrack) about the bug spray...the name'll come to me....
One More Kiss from Follies (not because it's so bad, but because the rest of the score is so great that the song just drags it down for me)
Good songs in bad shows:
Memory in Cats
About four songs each in Sunset Boulevard and Aspects of Love
Mama A Rainbow in Minnie's Boys (Man was I disappointed when I heard the rest of that score! There were maybe two other songs that were decent, the rest were pretty blah - but Mama a Rainbow is just WOW).
Hi, Shirley Temple Pudding.