Old English sliefe (West Saxon), slefe (Mercian) "arm-covering part of a garment," probably literally "that into which the arm slips," from Proto-Germanic *slaubjon (source also of Middle Low German sloven "to dress carelessly," Old High German sloufen "to put on or off"). Related to Old English slefan, sliefan "to slip on (clothes)" and slupan "to slip, glide," from PIE root *sleubh- "to slide, slip."
Compare slipper, Old English slefescoh "slipper," slip (n.2) "woman's garment," and expression slip into "dress in." Mechanical sense is attested from 1864. Meaning "the English Channel" translates French La Manche, literally "the sleeve" (from Old French manche "a sleeve," also "a handle," from Latin manicae "long sleeves of a tunic;" see manacle (n.)).
To have something up one's sleeve is recorded from c. 1500 (large sleeves formerly doubled as pockets); to have a card (or ace) up one's sleeve in the figurative sense "have a hidden resource" is from 1863; the cheat itself is mentioned by 1840s. To wear one's heart on (one's) sleeve is from "Othello" (1604).