Old English stream "a course of water," from Proto-Germanic *strauma- (source also of Old Saxon strom, Old Norse straumr, Danish strøm, Swedish ström, Norwegian straum, Old Frisian stram, Dutch stroom, Old High German stroum, German Strom "current, river"), from PIE root *sreu- "to flow."
From early 12c. as "anything issuing from a source and flowing continuously." Meaning "current in the sea" (as in Gulf Stream) is recorded from late 14c., as is the sense of "steady current in a river." Stream of consciousness in lit crit first recorded 1930, originally in psychology (1855). Stream of thought is from 1890.
early 13c., "to flow copiously," from stream (n.). Transitive sense "discharge in a stream" is from late 14c. Related: Streamed; streaming. Compare German strömen, Dutch stroomen, Danish strömme, all verbs from nouns.