1610s, "firmly, securely," from solid (adj.) + -ly (2). The meaning "unanimously" is from 1865, American English.
Entries linking to solidly
solid adj.
late 14c., "not empty or hollow, hardened;" of figures or bodies, "having three dimensions," from Old French solide "firm, dense, compact," from Latin solidus "firm, whole, undivided, entire," figuratively "sound, trustworthy, genuine," from suffixed form of PIE root *sol- "whole."
The meaning "firm, hard, compact" is from 1530s. Of arguments, etc., "substantial" (opposed to frivolous or flimsy). The meaning "entirely of the same stuff" is from 1710. Of qualities, "well-established, considerable" c. 1600. Of food from c. 1700.
As a mere intensifier, "thoroughly, downright," by 1830. The slang sense of "wonderful, remarkable" is attested by 1920 among jazz musicians.
As an adverb, "solidly, completely," 1650s. Solid South in U.S. political history is attested from 1858 on the notion of unanimity in voting; solid in this sense (in reference to New York) is by 1855. Solid state as a term in physics is recorded from 1953; the meaning "employing printed circuits and solid transistors" (as opposed to wires and vacuum tubes) is from 1959.
-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.