Well, nothing's ever going to be familiar to a "general audience" from most of these shows, but some could certainly become standards. I've tried to pare these down to "songs I think are good or excellent" and/or "songs which are already regarded as standards", although I'm undoubtedly being inconsistent with myself.
Trying to stick to Broadway only:
Sirens (JANE EYRE)
At the Fountain (SWEET SMELL OF SUCCESS)
Lot's Wife (CAROLINE OR CHANGE)
Astonishing (LITTLE WOMEN)
Here I Am, Dirty Rotten Number (DIRTY ROTTEN SCOUNDRELS)
A Way Back To Then (TITLE OF SHOW)
Sail Me Away (LESTAT)
Revolutionary Costume For Today, Another Winter in a Summer Town (GREY GARDENS)
I'll Be There (PIRATE QUEEN)
Who I'd Be (SHREK)
Model Behaviour, Invisible (WOMEN ON THE VERGE)
Fly Fly Away (CATCH ME IF YOU CAN)
With You (GHOST)
When It All Falls Down (CHAPLIN)
Raining (ROCKY)
I'm Gone (HANDS ON A HARDBODY)
To Build A Home, Another Life, Falling Into You, Before and After You/One Second and a Million Miles, It All Fades Away (BRIDGES OF MADISON COUNTY)
Always Starting Over (IF/THEN)
Anywhere But Here (HONEYMOON IN VEGAS)