bourn (n.1)
also bourne, "small stream," especially of the winter torrents of the chalk downs, Old English brunna, burna "brook, stream," from Proto-Germanic *brunnoz "spring, fountain" (cognates: Old High German brunno, Old Norse brunnr, Old Frisian burna, German Brunnen "fountain," Gothis brunna "well"), ultimately from PIE root *bhreue- "to boil, bubble, effervesce, burn" (see brew (v.)).
bourn (n.2)
"destination," 1520s, from French borne, apparently a variant of bodne (see bound (n.)). Used by Shakespeare in Hamlet's soliloquy (1602), from which it entered into English poetic speech. He meant it probably in the correct sense of "boundary," but it has been taken to mean "goal" (Wordsworth, Matthew Arnold) or sometimes "realm" (Keats).
The dread of something after death, The vndiscouered Countrey; from whose Borne No Traueller returnes. ["Hamlet" III.i.79]