The arduino requires a 5V regulated power supply. Using a trick I learned right here on Instructables, I cracked open an old car-cigarette lighter type phone charger & pulled out the circuit board. I hooked it up to my 11.1V battery on the cigarette lighter side and the arduino on the phone side. Be sure to test the voltage output from the charger with a multimeter to make sure you're really getting about 5V, significantly more could fry your arduino. I added a tiny Lilypad slide switch between the power supply and the battery to allow me to turn the umbrella on & off.
I sewed a holster for the battery using old nylon strap. Velcro allows me to open and close the holster to replace the battery. I then zip-tied the holster to inside top of the umbrella around a rib and the main tube.
Finally, sewed the arduino and the power supply to the umbrella and hooked up all the wires.