Old English lætan "to allow to remain; let go, leave, depart from; leave undone; to allow; bequeath," also "to rent" (class VII strong verb; past tense let, past participle læten), from Proto-Germanic *letan (cf. Old Saxon latan, Old Frisian leta, Dutch laten, German lassen, Gothic letan "to leave, let"), from PIE *le- "to let go, slacken" (cf. Latin lassus "faint, weary," Lithuanian leisti "to let, to let loose;" see lenient). If that derivation is correct, the primary sense would be "let go through weariness, neglect."
Of blood, from late Old English. To let (something) slip originally (1520s) was a reference to hounds on a leash; figurative use from 1540s. To let (someone) off "allow to go unpunished" is from 1814. To let on "reveal, divulge" is from 1725; to let up "cease, stop" is from 1787. Let alone "not to mention" is from 1812.