If you have an early morning departure or a late evening arrival and don't want to travel to/from the city at night, take advantage of the hotels close to the airport.
Hotels near the airport
Right on the airport grounds, just about a 3-minute walk from the terminal, you can stay in a hotel ibis Styles Budapest Airport with prices around 92 eur per night in a room for 2.
Hotels in Budapest
Near the nearby Köbanya-Kispest metro station, from which the 200E bus runs 24 hours a day (the journey takes about 22 minutes), you can stay at hotelu Chesscom for prices from 33 eur/night/2bedroom. Just note that at night the 200E bus does not leave from the bus station, but from the stop at the P+R car park.