The name comes from a passage in the Bible - 1 Kings 19. This chapter describes how God provides rest for His servant Elijah after his wearying battle with the false prophets (and Ahab & Jezebel) on Mt. Carmel. Read this great passage below.
The Story of the Broom Tree from 1 Kings 19:1-8 (NIV)
Now Ahab told Jezebel everything Elijah had done and how he had killed all the prophets (of Baal) with the sword.
So Jezebel sent a messenger to Elijah to say, "May the gods deal with me, be it ever so severely, if by this time tomorrow I do not make your life like that of one of them." Elijah was afraid and ran for his life. When he came to Beersheba in Judah, he left his servant there, while he himself went a day's journey into the desert. He came to a broom tree, sat down under it and prayed that he might die. "I have had enough, LORD," he said. "Take my life; I am no better than my ancestors."
Then he lay down under the tree and fell asleep. All at once an angel touched him and said, "Get up and eat." He looked around, and there by his head was a cake of bread baked over hot coals, and a jar of water. He ate and drank and then lay down again. The angel of the LORD came back a second time and touched him and said, "Get up and eat, for the journey is too much for you." So he got up and ate and drank. Strengthened by that food, he traveled forty days and forty nights until he reached Horeb, the mountain of God.