From the experts:
Mari enjoyed some mid-century popularity as a nicknameish version of Mary, but now deserves a second look as a multicultural classic. Mari is the name of the Basque goddess of fertility and weather and is also, for unrelated reasons, popular in Norway. The biggest problem with the potentially-lovely Mari will be getting people to say it the way you want. While some Europeans pronounce it as Marie, that will only confuse things more: Better to make it its own individual name by pronouncing it MAH-ree. As one commenter pointed out, in Estonian the name is not a Mary variation but means "berry" -- all the more reason for us to like it!