late 15c., "singly, alone, only," from sole (adj.) + -ly (2). Hence "exclusively" but also "entirely."
Entries linking to solely
sole adj.
"single, alone in its kind; one and only, singular, unique; having no husband or wife, in an unmarried state; celibate," late 14c., from Old French soul "only, alone, just," from Latin solus "alone, only, single, sole; forsaken; extraordinary," a word of unknown origin, perhaps related to se "oneself," from PIE reflexive root *swo- (for which see so).
-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.