"showy, tastelessly rich," c. 1600; earlier "joyfully festive" (1580s), probably a re-adjectivizing of gaudy (n.) "large, ornamental bead in a rosary" (early 14c.) via the noun gaud + -y (2.). In early Modern English it also could mean "full of trickery" (1520s).
Or possibly the adjective is from or influenced by Middle English noun gaudegrene (early 14c.), name of a yellowish-green color or pigment, originally of dye obtained from the weld plant (see weld (n.1)). This Germanic plant-name became gaude in Old French, and thus the Middle English word. Under this theory, the sense shifted from "weld-dye" to "bright ornamentation."
As a noun, "feast, festival" 1650s, from gaudy day "day of rejoicing" (1560s).
-ly 2
common adverbial suffix, forming from adjectives adverbs signifying "in a manner denoted by" the adjective, Middle English, from Old English -lice, from Proto-Germanic *-liko- (cognates: Old Frisian -like, Old Saxon -liko, Dutch -lijk, Old High German -licho, German -lich, Old Norse -liga, Gothic -leiko); see -ly (1). Cognate with lich, and identical with like (adj.).
Weekley notes as "curious" that Germanic uses a word essentially meaning "body" for the adverbial formation, while Romanic uses one meaning "mind" (as in French constamment from Latin constanti mente). The modern English form emerged in late Middle English, probably from influence of Old Norse -liga.