Hermione isn't a Muggle-born... she's a Half-blood. But that's not it, she's also half veela.
Hermione's mother, Samantha from a long line of witches and veelas, went to Beauxbatons for witches, and had a odd fascination for muggles, mostly their teeth. So after she finished school Sam went off to a college to become a dentist in England, and there, she met Darrin. He was handsome, charming, smart, and most importantly, a muggle. Love bloomed, and soon there were wedding bells ringing in the nearby church for the couple, but Sam could not give up her most important secret, that she was a witch. Two years passed and Samantha was pregnant and she was terrified of what Darrin would think when he found out that his wife and child were magical, and had an ability to attract their opposite gender to themselves. The child could be a squib, but Sam knew that that was almost impossible, she would have felt it, and besides the kid would still be a veela. So one night, before the baby was born, Sam sat Darrin down and told him what she was.
"Darrin, darling, there's something I must tell you, and I don't think that you'll like it I-
"Is it the baby? OH GOD! IT IS ISN'T IT? I knew we shouldn't have gone to that stupid wedding! What were Marty and Jay thinking, giving you that champagne! WHAT WAS I THINKING LETTING THEM? We should go over to them and-"
"We aren't going to do anything to your brothers. The baby's fine." Sam said calmly. Dan realized that a bomb would go off if he said anything else, so he motioned for her to continue. What can I say? He somewhat understands women!
"Darrin, I'm a witch, a broom-riding, house-haunting, cauldron-stirring witch! And so is our child, though the baby can be a wizard too. And I'm a vee-"
"WHAT?"