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.
"stoppage, obstruction" (obsolete unless in legal contracts), late 12c., from archaic verb letten "to hinder," from Old English lettan "hinder, delay," from Proto-Germanic *latjanan (cf. Old Saxon lettian "to hinder," Old Norse letja "to hold back," Old High German lezzen "to stop, check," Gothic latjan "to hinder, make late," Old English læt "sluggish, slow, late"); see late.