Well how about a "morning set" at an authentic salaryman kissaten. This one opens at 7:00 and is on the 2nd floor of the World Trade Center building - connected to the Monorail station. Coffee, a thick slice of white toast, and some salad for 390 yen.
http://www.meijiparlor.co.jp/temps/break.html

That's an authentic Japanese experience of sorts.

After breakfast, you could walk to Zojo ji temple. Opens at 9:00 am. I wouldn't bother going up Tokyo Tower, but you'll see it there.

Or ... My splurge recommendation ... Get in a cab at Haneda and go straight to the Tokyo Oedo Onsen Monogatari - Hot springs on O-Daiba island. Last entry is at 7:00 am, so you need to get your skates on. But once there, you can soak in a tub for a couple of hours. Depending on your level of confidence, you could try taking public transport back to Haneda Airport.