I, like you, thought the version I heard on the radio was Cyndi Lauper singing "Santa Baby" so I did a bunch of searches. Cyndi did just sing the song with Rosie O'Donnell at the Cyndi & Friends Home for the Holidays show in New York but the version that I'm positive you're thinking of is actually Madonna who recorded it in the later part of the 80s for a celebrity charity Christmas album that was never released. I really thought it sounded a lot like Cyndi too but it turns out it isn't. The voice you hear in the video is Madonna: http://youtu.be/5ycWObpi73Y
It was Cyndi Lauper. I had a Christmas cassette (yes, I'm THAT old) and Cyndi's version of Santa Baby was on it, along with other Christmas songs by other artists. It was fantastic -- she is the queen of sassiness mixed with a good voice. But I don't have the cassette anymore and do not remember the name of the collection, darn it. I've been looking for two years for it.