Budapest is big but the useful bases are small, and the choice is mostly about how much night noise you are willing to sleep above and which side of the river you want to wake up on.
District V — Belváros and Lipótváros
The riverside core between the Chain Bridge and Parliament: the Danube promenade, the Basilica and the main interchanges are all walkable, and three metro lines meet at Deák Ferenc tér. It is the most expensive district to sleep and eat in, and it empties of residents at night.
District VII — Erzsébetváros, the Jewish Quarter
Ten minutes' walk from the centre and the densest concentration of bars, hostels and late food in the city, so beds are cheap for the location. The trade-off is that Kazinczy and Akácfa utca run until dawn most weekends — ask for a courtyard-facing room.
Palace Quarter and Ferencváros — Districts VIII and IX
The blocks behind the National Museum, around Ráday utca and the Corvin quarter: nineteenth-century apartment houses, students, cheaper flats and ordinary neighbourhood restaurants. Metro M3 and tram 4/6 put you in the centre in about ten minutes.
Buda side — Districts I, II and XI
Quieter and greener, with Castle Hill for the view and Újbuda around Bartók Béla út for everyday prices. You will cross a bridge every day, so pick somewhere on tram 4/6, tram 19/41 or the M4 rather than halfway up a hill.