Where to stay gets asked enough, and I keep giving my probably biased answer, so I decided to think it through a bit more.
District 1 – Buda:
The castle district portion of District I, if you have been to Budapest a number of times, love Budapest’s District V, and just want to do something similar but different. You will be a 30-minute bus trip (or a longer walk) to the overwhelming majority of the tourist sights, as they all exist in Pest. The Buda river embankment for the views and a change of scenery, new views across the river, but only if you have done Budapest a few times. Again, more akin to District V than anything else. From here you can reach Pest in about 15 minutes by tram.
District V – The River Boat Tourist District:
If you love Venice, then you will maybe love District V. It’s the most fully restored part of town. It’s the most fully developed for tourism. The setting is quite beautiful. The prices are the highest by a fairly substantial degree. The views from the Intercontinental or the Marriott are second to none in Europe. The ambiance along the river front (the Corso) is second to none in Europe. If you have a short stay, then this is the place to be despite the “tourism factor”. Reaching the major sights in Pest is quick and easy from here as well.
District VI – The Glory of Andrassy ut and Pest as it was in 1920:
The Champs-Élysées in Paris wishes it were as beautiful and interesting as Andrassy ut; and Andrassy ut runs up the center of District VI. Move a few blocks off Andrassy ut and find yourself in Pest as it was 100 years ago. Somewhat unrestored, full of shops and restaurants and bars and endless streets of magnificent architecture housing the citizens of Pest today, just as they did 100 years ago. For me this is vibrant Budapest at its best. Stay between the inner and outer loop road and vast majority of the sights of Pest are within a 10 minute trip.
District VII – Jewish Budapest and Party Central
You cant know Budapest if you don’t know the Jewish experience. So this is a place you must visit. During the day it is a fascinating district. At night, well, if you are 20 years old and came to drink and party, this is indeed the place to be after 9pm. Otherwise not a great place to stay.
District VIII – The mansions of Budapest.
For 10 years I have heard that this is the up and coming district of Budapest, but that just hasn’t emerged from a perspective of tourism yet. Still, a great place to explore for museums and architecture on your 5th day in Budapest. Because of some limitations in public transport (you will be a bus a lot) not the most convenient for a first time tourist.