Harriet has long been considered a stylish, upscale name in England, but it's still waiting to be revived in the US—though some parents seeking a solid, serious semi-classic are beginning to consider it.
Distinguished namesakes include Harriet Beecher Stowe, author of the influential Uncle Tom's Cabin; Harriet Tubman (born Araminta), the famed abolitionist; and in pop culture, Harriet Hilliard Nelson, of Ozzie & Harriet TV fame. Many will associate it with the protagonist of the childhood classic Harriet the Spy. Harriet has appeared in other books ranging from Jane Austen's Emma to Girl with the Dragon Tattoo.
Harriet, a Top 100 name at the turn of the last century, hasn't appeared on the list since 1970. On the other hand, it is currently Number 61 in England and Wales.
Cutesy nicknames for Harriet include Hattie, Hetty, and Hatsy. A related idea: Henrietta.