'stably' is formed in English from the adjective 'stable' plus the adverbial suffix '-ly'. 'Stable' ultimately comes from Latin 'stabilis', where 'stare' meant 'to stand'.

'stable' developed from Latin 'stabilis' and related Latin 'stabulum' (a standing place), passed into Old French as 'estable' and into Middle English as 'stable', and the adverb 'stably' was formed later by adding '-ly' to the adjective.

Initially related to 'standing' or a 'standing place' (for the noun) and to 'steadfast' (for the adjective); over time the adjective kept the sense of 'firm/steady' and the adverb came to mean 'in a firm or unchanging manner'.