This used to be my favorite name when I was 8 and it's starting to grow on me again. Back then my parents told me that it sounded too middle-aged but I thought I'd ask you berries? Is it due for a comeback? What about with the nicknames Janie and Nettie (any other ideas)?
Although I would associate it with middle aged women, Janet could work on a little girl with the nicknames Janie and Nettie (cute!). It has an interesting late-retro imagery.
I, too, loved the name Janet as a child. I heard it on an episode of Scooby Doo and practically fell in love with it. I told my mom I thought the name was pretty and her reply was more or less the same as your mother's response. I agree, it is ready for a comeback. I never saw it as "dated" but mature, pretty, and classy. Nicknames that I can think of would by Jan. Something simple. Hope I helped!