Is the wife obligated to inform her father that she met her husband through Facebook, or is what she did by not informing him permissible?

The girl did well by not informing her father about meeting the young man via Facebook, as this is not a religious obligation upon her. If her father knew, he would not have the right to refuse the marriage merely because of the method of acquaintance, for the criteria for choosing spouses are religion and character. Conversation between a young man and a young woman over the internet is permissible when there is a need and within the bounds of Islamic etiquette. The girl was correct if she used dissimulation in her speech so as not to fall into lying.