Arduino WiFi enabled LED
Based off of your two specifications of

Battery powered (so all items must use minimum power)
WiFi enabled

I would recommend not going with an Arduino at all as it will be much more difficult to meet the minimum power requirement. In order to minimize your power you will need to look into spinning your own board with a very low power microcontroller. It seems like almost every microcontroller company sells some version of ultra low power, but the MSP430 seems to be the most widely used line for low power.
As far as the WiFi enabled part, this is a little bit more difficult to get to be low powered. If you are fine with having your device waking up to check in to see what it needs to do with the LED then you can get your power even lower. However, my guess is that you are actually wanting to be able to send your device a packet at any time to tell it to do something with the LED. In order to go this path with lowest power you would be best rolling your own WiFi solution, but my guess also is that you do not care to do this. So you will want to look for a module that will do this for you. Roving Networks has a few options that support 4uA sleep mode and 38mA active.
With all of that said, if you still want to go with an Arduino (even though you didn't say it was a requirement) you will have to settle for very poor battery life. By very poor battery life I mean something less then a day unless you want to lug around a car battery.