Dag = day and mar or mare = maiden, girl so the name means "maiden of the light of day." The Bohemian noblewoman who became Queen Dagmar of Denmark was popular. She died in childbirth. In popular ballads she was revered like a saint known for goodness and mercy. She was the consort of Valdemar II. He married Bengaria next, who was known for her beauty and ambition, even though she had made him promise not to do this. So her story also has a streak of martyrdom: Valdemar was campaigning when she lay dying, and so she was seen as abandoned. And then he betrayed her memory by remarrying against her wishes. The name in Danish is pronounced 'Dao-mar' with the accent on the first syllable.
My mother's name is "Dagmar" and she does not credit her name at all, unfortunately. It comes from the Germanic words "dag" = "day" and "mar" ("mor"?) = "big, great", so "Dagmar" would be "great day".