Originally Posted by Misco60

This isn't correct. The TXL bus starts at Alexanderplatz and goes to Tegel via Hauptbahnhof (main station). The express bus from Zoo is the X9.

Depending on where you are in town, the fastest way to the airport is often to take the U6 north to Kurt-Schumacher-Platz and then bus 128 directly outside the station (follow the signs on the platform). The TXL and (to a lesser extent) X9 airport buses can get very crowded and, depending on the time of day, get stuck in traffic.

I would definitely echo the advice to see your mother off for her flight and return to Berlin for the rest of the day. The lounge at Tegel is small and dull.

Enjoy your time in this amazing city! I've lived here since June and love it.