I'm always making list of '60s actresses who should have become stars, but why limit myself to the '60s? A starlet of the '40s and '50s whom I always thought should have had a bigger career was Janis Paige. (It seems a bit random for me to be talking about her, but I was just re-watching Silk Stockings, where she plays a supporting part and almost steals the movie from Fred Astaire and Cyd Charisse.) She was a likable performer and exceptionally good-looking; she could be a heroine, a vamp, or a wisecracking best friend. She also had an adequate singing voice (though she suffered from pitch problems sometimes) that allowed her to do well in musicals, like the original Broadway production of The Pajama Game. For some reason she never made the leap to leading-lady status on the screen, maybe because her versatility worked against her -- she could play heroines, but she didn't have to, and when an actress doesn't clearly fit into one particular category, the studios don't always know how to cast her.
This page has a good biographical piece on Paige.