I loled greatly at the 'got it from Shakespeare actually' thing. I mean, JK Rowling is a well-respected, award-winning children's author, and the
Hermione in her books - named for the Shakespeare character! - is famous for her BRAIN. It's not as if we're talking a name popularised by some cheesy hack on a heroine famous for multiple terrible porn scenes. And, as others have pointed out, there are other Hermiones about; it's not as if it's completely unheard of outside the HP books.
I like it a lot, but really don't like any of the potential short forms, which would deter me from using it.
Hermione Beatrice works well for me; these do too:
Hermione AugustaHermione OctaviaHermione ConstanceHermione LucyHermione ClareHermione AmandaHermione LouisaHermione LilacHermione JoanHermione IsabelHermione JuliaHermione MayHermione MaudHermione Sybil
Also... you could always go for
Hermia, if you want a Shakespearean name unsullied by plebeian associations! :p