Old English bledan, "to cause to lose blood, to let blood" (in Middle English and after, especially "to let blood from surgically"), also (intrans.) "to emit blood," from Proto-Germanic *blodjan "emit blood" (source also of Old Norse blæða, Dutch bloeden, German bluten), from PIE *bhlo-to- "swell, gush, spurt," or "that which bursts out," from suffixed form of root *bhel- (3) "to thrive, bloom."
Meaning "extort money from" is from 1670s. Of dyes or paints, "to be washed out," from 1862. Figuratively, of the heart, "to suffer anguish, feel pity or sorrow," late 14c.