slight (adj.)
early 14c., "flat, smooth; hairless," probably from a Scandinavian source akin to Old Norse slettr "smooth, sleek," from Proto-Germanic *slikhtaz (cognates: Old Saxon slicht; Low German slicht "smooth, plain common;" Old English -sliht "level," attested in eorðslihtes "level with the ground;" Old Frisian sliucht "smooth, slight," Middle Dutch sleht "even, plain," Old High German sleht, Gothic slaihts "smooth"), probably from a collateral form of PIE *sleig- "to smooth, glide, be muddy," from root *(s)lei- "slimy" (see slime (n.)).

Sense evolution probably is from "smooth" (c.1300), to "slim, slender; of light texture," hence "not good or strong; insubstantial, trifling, inferior, insignificant" (early 14c.). Meaning "small in amount" is from 1520s. Sense of German cognate schlecht developed from "smooth, plain, simple" to "bad, mean, base," and as it did it was replaced in the original senses by schlicht, a back-formation from schlichten "to smooth, to plane," a derivative of schlecht in the old sense [Klein].
slight (v.)
c.1300, "make plain or smooth," from slight (adj.) Meaning "treat with indifference" (1590s) is from the adjective in sense of "having little worth." Related: Slighted; slighting.
slight (n.)
1550s, "small amount or weight," from slight (v.). Meaning "act of intentional neglect or ignoring out of displeasure or contempt" is from 1701, probably via 17c. phrase make a slight of (1610s).