A lease buyout is the purchase of a leased vehicle by the person leasing it, at a price the lease itself sets. The federal consumer leasing rules call the right to do this a purchase option. 12 CFR 1013.4(i), in Regulation M, requires a lease to disclose "A statement of whether or not the lessee has the option to purchase the leased property", and then, if the option exists at the end of the lease term, "the purchase price", or if it may be exercised earlier, "the purchase price or the method for determining the price and when the lessee may exercise this option."
The phrase "lease buyout" is what people search for and what dealers say. It is also used for two other things that are not this: a loan taken out to pay the option price, which is ordinary closed-end credit, and a landlord or tenant paying to end a property lease early, which shares nothing with this subject but the words. Published material on car leases covers the lease itself; this page covers the decision at the end of it.