'letting' ultimately comes from Old English via the verb 'lǣtan' (also seen as 'lætan'), meaning 'to allow, to leave, to let'.

'lǣtan' (Old English) passed into Middle English as 'leten'/'let' and developed the noun form 'letting' (the act of letting or allowing); by Early Modern English the sense 'to let out for payment, to hire out' became established and gave rise to the property-related noun 'letting'.

Initially it meant 'to allow, to leave alone' or more generally 'to permit'; over time a specialized meaning 'to give the use of something to someone in return for payment (to rent out)' emerged and is now common in contexts like property and housing.