Frequently found within the burgeoning ranks of the insomniacs, I usually go downstairs and reload ammo, clean weapons, design new kites, work on a new electronics package, along with some DX (distance listening to my shortwave radio) or fire up the Ham Shack. I may also practice my Morse Code. Sometimes I go goodies shopping at one of the 24hr supermarkets open near my house.
In years past, I might go to the finish of one or another of my favorite twisty roads, scout the road surface, cleaning gravel and rockfalls - this duty sometimes took hours, by which time the sun had risen.
Some book stores or newsstands are open early, or are 24 hr operations in big cities - absent the press of humanity, they are fine places to while away a few hours.
If I wake up really early, I check the ephemeris for what might be interesting to see, then look for good sky - if it is clear, then I pull out the Celestron, and wander the bejewelled depths of the universe.
A walk or drive through the neighborhood is restful and informing, and works as an informal crimewatch. I make one or two "busts" a year doing this - usually some teenaged and/or alcohol/drug fueled mischief.
Sometimes I plan, shop, and cook a gourmet breakfast for my Lady.