late 15c., "to take a lease," from Anglo-French lesser (13c.), Old French laissier "to let, let go, let out, leave" "to let, allow, permit; bequeath, leave," from Latin laxare "loosen, open, make wide," from laxus "loose" (from PIE root *sleg- "be slack, be languid"). Medial -x- in Latin tends to become -ss- or -s- in French (compare cuisse from coxa). The Latin verb also is the source of Spanish laxar; Italian lasciare "leave," lassare "loosen."
Compare release (v.). Meaning "to grant the temporary possession of at a fixed rate" is from 1560s. Related: Leased; leasing. The form has been influenced by the noun, and the modern sense of "to take a lease" might be a new 19c. formation. Lessor, lessee in contract language preserve the Anglo-French vowel.