Stay in Berlin for one night near Tegel airport

I would recommend Meininger Berlin Central Station. It is half-hotel half-hostel, so you can find good rooms and facilities for low prices. I've stayed there twice in their hostel style 4-bed rooms. Nice, clean, and no problems. It is literally next to the fairly new Berlin Central rail station, which connects you to all the major parts of the city (and other parts of the country or other countries) by train, and also has many bus routes that include an express service to both airports. It is also in reasonable walking distance of the major tourist attractions.

There is also the Sunflower Hostel that is near Warschauer Str. It's not exactly central, but near public transport, and last time I took a short walk from Berlin Ost station. If you cross Warschauer Str. from there, you end up in the heart of Friedrichshain, with lots of restaurants and bars.