Connectiong the "peaceful" interpretation that aren't related to etymology,I can hazard two guesses.
First, Lime in Europe refers to the Tilia family of trees, otherwise known as Lindens (and Basswood in North America). The flowers and fruit of Linden trees contain a medicinal compound that calms and soothes and has a pleasant taste and odor. The flower has a similiar smell to citrus lime flower but they are unrelated plants.
In European history, lime trees were often symbols of love and peace. Probably because they were so common, smelled good, and provided an excellent source of nectar for honey bees. There's a famous street in Germany that is lined with lindens---in Berlin I think, but I'm not up to research right now.