Old English brom, popular name for several types of shrubs common throughout Europe (used medicinally and for fuel) and characterized by long, slender branches and many yellow flowers, from Proto-Germanic *bræmaz "thorny bush" (source also of Dutch braam, German Brombeere "blackberry"), from PIE *bh(e)rem- "to project; a point."
As "twigs of broom tied together to a handle to make a tool for sweeping," mid-14c. Traditionally, both the flowers and sweeping with broom twigs were considered unlucky in May (Suffolk, Sussex, Wiltshire, etc.).