This is my niece's name and is pronounced the same as Dana {DAY-nə}. Her mom choose to spell it this way as it looked more feminine and softer then Dana. Personally, I've always loved her name spelled this way. However, I know that she gets a lot of people trying to call her Diana {die-AN-ə}. But, she's used it for almost two decades, and I haven't ever heard her complain about it either!
Daina is a traditional name for folk music in the Baltic languages, preserved in Lithuania and Latvia. Lithuanian dainas (literally, "songs") are often noted not only for their mythological content, but also for preserved historical messages.