[ Etymonline on 'etui (n.)' : ] 1610s, also ettuy, etwee from French étui, Old French estui (12c.) "case, box, container," back-formation from
estuier "put in put aside, spare; to keep, shut up, imprison," which is of uncertain origin.
Perhaps from Latin studere "to be diligent."
[ Wiktionary : ] From Old French estui, from estuier (“keep, hold”), itself possibly
from Vulgar Latin *studiāre, from Latin studium.
Both etymologies hedge with adverbs to communicate their uncertainty about the semantic shift from Latin to Old French, but what might explain the semantic shift?