Green refuses to be one story. Dictionary of American Family Names (via Ancestry) and FamilySearch both describe English Green as either a nickname—someone in green cloth, a young “green” person, or a May-game “Green Man”—or a place name for someone living by the village green.
Both paths are real; different families took different doors into the same spelling. Immigrant Grün or Groen sometimes became Green in America too. Lead with honesty: your Green may be the grassy common, the green coat, or another language’s “green” name. That uncertainty is the accurate family tale, not a failure of research.