Are any Arduino boards especially suited for low-power operation? The Arduino project has done an excellent job in providing a portable software interface to a number of different hardware systems.

Are there any Arduino-compatible systems which are especially applicable for very low power battery-based operation?  Specifically, in an application where a device wakes up periodically (e.g. for data collection) or activated by a user button press?
Technically you can use any arduino for that purpose with some modifications. I recommend to go with an arduino Pro Mini and remove the LED and the voltage regulator. 
But if you don't need so much I/O you can go with a digispark Attiny board.