What fifthtime said; pretty names, but sometimes tragic or bloody myths attached. Here are some Greek names ending in the same sound as Hermione, but would be a good idea to check them out - and also their correct pronunciation - if you really want to use them. Splendid site for some of the mthological ones here: www.theoi.com/
Ariadne, Cybele, Hecate, Amphitrite, Tyche, Selene, Persephone, Ismene, Asterope, Merope, Kyllene, Aphrodite, Berenike, Ianthe, Xanthippe...
and also, of course, Daphne, Penelope, Phoebe, Irene, Thisbe (as in Midsummer Night's Dream)
And Astarte (though not of Greek origin)