Personally, I believe that Anitra was a Norwegian playwright's idea of what an Arabic name ought to sound like. It is not truly a Danish name, or an Arabic name, or a given name in any other culture I have been able to find, before the time of Ibsen's play.
One possible inspiration for the name: there is a tiny town name "Enitra" near where Peer Gynt would have crossed the border into Arabia at the time of his story.
It is, whatever, a lovely name. It does seem to be a variant of the eternal Ann (Anna, Anita, Annabelle, etc.) which traditionally, in Christian cultures, means "beloved of God".
The Anitra that Peer Gynt met, and others in literature and history, may stretch that meaning a bit, but most of them are colorful characters.
The Arabic princess Anitra first met Peer Gynt when he was exceedingly rich with the wealth he had won from solving the riddle of the Sphinx. She charmed him, danced for him, cooed him into giving her all of his wealth, and then she told him to scarper off before she sicced Daddy's guards on him.
A legendary gypsy dancer named Anitra was said to be able to charm audiences so thoroughly that her relatives could walk among them and steal the contents of their pockets with impunity.
Arthur C. Clarke used the name Anitra for a character in his novel The Fountains of Paradise. Anitra was the retarded infant daughter of the main character; but at least he used the name.
Anitra was the hawk belonging to one of the central characters in the novel Windhaven by George R. Martin and Lisa Tuttle.
Somewhere out there is a rose named Anitra. One day many years ago I was talking with a friend in the halls of the Seattle YWCA when a woman waving a paintbrush came bursting out of a nearby room. "I've just been painting a picture of a rose. Can I call it Anitra's Rose?" I thought it was a very lovely rose and was honored to have it named after me. If the artist ever reads this, could you let me know whatever became of that rose?
I'm hoping that other people will add to the stories I have collected here of the name Anitra. This is fun. I have also had several Anitras sign my guestbook over the years, and I've found others in the search engines -- or when I tried to sign up for a service and found they already had several Anitras! I've added everyone I've found to the Anitra List, and I hope you add yourself if I haven't found you yet.